# The Drift of #MyBodyMyChoice Discourse on Twitter

Cristina Menghini  Brown UniversityUSA<EMAIL_ADDRESS>,  Justin
Uhr  Brown UniversityUSA<EMAIL_ADDRESS>,  Shahrzad Haddadan  Brown
UniversityUSA<EMAIL_ADDRESS>,  Ashley Champagne  Brown
UniversityUSA<EMAIL_ADDRESS>,  Björn Sandstede  Brown
UniversityUSA<EMAIL_ADDRESS>and  Sohini Ramachandran  Brown
UniversityUSA<EMAIL_ADDRESS>
(2022)

###### Abstract.

#MyBodyMyChoice is a well-known hashtag originally created to advocate for
women’s rights, often used in discourse about abortion and bodily autonomy.
The Covid-19 outbreak prompted governments to take containment measures such
as vaccination campaigns and mask mandates. Population groups opposed to such
measures started to use the slogan “My Body My Choice” to claim their bodily
autonomy. In this paper, we investigate whether the discourse around the
hashtag #MyBodyMyChoice on Twitter changed its usage after the Covid-19
outbreak. We observe that the conversation around the hashtag changed in two
ways. First, semantically, the hashtag #MyBodyMyChoice drifted towards
conversations around Covid-19, especially in messages opposed to containment
measures. Second, while before the pandemic users used to share content
produced by experts and authorities, after Covid-19 the users’ attention has
shifted towards individuals.

twitter, covid, topic change, hashtags

††copyright: acmcopyright††journalyear: 2022††copyright:
acmlicensed††conference: 14th ACM Web Science Conference 2022; June 26–29,
2022; Barcelona, Spain††booktitle: 14th ACM Web Science Conference 2022
(WebSci ’22), June 26–29, 2022, Barcelona, Spain††price: 15.00††doi:
10.1145/3501247.3531570††isbn: 978-1-4503-9191-7/22/06

##  1\. Introduction

Many factors impact what and how users write on online social networks (OSNs).
These factors include platform-level changes such as when Twitter changed the
number of characters per tweet from 140 to 240 in 2017 (Gligorić et al.,
2018), external events such as a presidential election (Enli, 2017; Yaqub et
al., 2020), social movements such as the Arab spring or the Black Lives Matter
movement (Starbird and Palen, 2012; Ince et al., 2017), or crises such as
earthquakes or terrorist attacks (Mendoza et al., 2010; Cassa et al., 2013).
In this paper, we focus on how users adopt hashtags for a different purpose
than originally intended or _hijack_ hashtags. Understanding how users hijack
hashtags is important for tracking and predicting social movements within
public discourse.

Figure 1. Word cloud of the hashtags that co-occur with #MyBodyMyChoice in the
entire stream of tweets from 1-Jan-2018 to 20-Dec-2021.

The global health crisis resulting from the Covid-19 pandemic has impacted
many aspects of daily life, ranging from health to economics, and has changed
in certain ways how users behave on OSNs. Researchers have shown the role of
OSNs in the distribution of false news (Vosoughi et al., 2018), in part
leading to our current _infodemic_. Many researchers have studied the effect
of Covid-19 on Twitter, which is one of the most potent online platforms for
distribution of information. Some research shows that during the Covid-19
pandemic, user behavior changed on Twitter and other platforms, such as
Wikipedia (Ribeiro et al., 2021; Ruprechter et al., 2021). Gligorić et al.
(2020) shows that Twitter users’ attention shifted towards accounts associated
to healthcare, science, government and politics as opposed to accounts related
to religion or sports. As the impact of OSNs for spreading hate speech and
false news have raised concern, it is vital that online social network
administrators concentrate their efforts on excluding organized hate speech or
the spread of false news. A drift in the usage of a hashtag can make it more
difficult to identify false news or hate speech (Darius and Stephany, 2019,
2021).

On Twitter, discussions and topics are organized by usage of hashtags. When
users adopt a hashtag that already exists for another purpose than originally
intended, this is often called “hashjacking” or referred to as hashtag drift
or co-option. Researchers have studied hashjackings through a _linguistic
perspective_ , by analyzing the tweet itself, and through a focus on _user
behaviours_ , by researching how user groups interact with tweets (Hayashi et
al., 2015; VanDam and Tan, 2016; Darius and Stephany, 2019; Alfifi et al.,
2018; Xiong et al., 2019) (see Section 3.3 for a full discussion).

In this work, we examine the hashtag #MyBodyMyChoice (Figure 1) to see how the
content of the tweets associated with it evolved during the pandemic. The
hashtag originated to support women’s rights around abortion and bodily
autonomy, but during the pandemic a separate set of users utilized the hashtag
to communicate how vaccine mandates related to Covid-19 restricted their
bodily autonomy.

Our study shows that the hashtag #MyBodyMyChoice can no longer be uniquely
associated to the original discourse centered around women’s rights. Indeed,
after the Covid-19 outbreak there is an increasing number of monthly tweets,
up to 75%, related to _covid & vaccines_. Although the discourse about
#MyBodyMyChoice branches out into several directions, the characterization of
users interactions on the corpora of tweets related to _women’s rights &
abortion_ and _covid & vaccines_ suggests that the topic drift was not
intentionally planned by a group of users. In Section 3.3, we provide a
taxonomy of hashjackings including (1) semantic hashjacking; (2) hashjacking
with polarization intents; (3) hashjacking to capitalize on the success of the
original hashtag. We associate the case #MyBodyMyChoice to a semantic
hashjacking, which involves a natural topic shift.

Figure 2. Hashtags in our reference windows. The plots show the most occurring
hashtags appearing in tweets from BP, IP, and CC, from left to right. The
larger and darker a word is, the more often it appears during the time window.
Colors correspond to the respective topics: pink to _women rights & abortion_,
green to _covid & vaccination_, light blue to _politics & civil rights_, and
orange to _freedom of choice_.

##  2\. Data

We collected the stream of #MyBodyMyChoice tweets from 1-Jan-2018 to
20-Dec-2021. For the purpose of the analysis, we break down the stream
$S_{\texttt{\\#MyBodyMyChoice}}$ into three equal-in-length periods to study
the content of the hashtag before and after the Covid-19 outbreak. The three
reference windows are: (1) Before pandemic (BP) from 1-Jan-2018 to the first
officially reported case of Covid-19 outside China on 13-Jan-2020. (2) Initial
pandemic (IP) from 13-Jan-2020 to the beginning of the vaccination campaign
with the approval of Moderna vaccine by the FDA on 17-Dec-2020. (3) Covid-19
coexistence (CC) from 17-Dec-2020 to 20-Dec-2021.

Among the tweets in the stream, we retain the around 280K tweets written in
English, corresponding to 86% of the entire retrieved corpus (excluding tweets
from unidentified languages that are too short or contain too many hashtags to
infer the language). It follows that throughout the analysis we assume that
the content of the tweets mostly characterizes the ongoing debates in English-
speaking countries.

###  2.1. Data Sub-topics

In this section, we explain how we identify the main sub-topics of the
#MyBodyMyChoice hashtag, and how we assign a tweet to the respective sub-
topic. Particularly, given the stream $S_{\texttt{\\#MyBodyMyChoice}}$, we
call a _sub-topic_ the topic characterizing the content of a subset of the
tweets in the stream.

Identify sub-topics. To characterize the presence of sub-topics we use
hashtags. We refrain from the usage of topic-modeling algorithms for two main
reasons: (1) In preliminary analysis, we found the hashtags to be
representative of the ongoing discussions. Indeed, we observed that the text
of tweets about either abortion or Covid-19 vaccination might contain the same
set of words and differ only by hashtags: two examples are “I have the right
to choose #MyBodyMyChoice #reproductiverights” versus “I have the right to
choose #MyBodyMyChoice #novaccine”. (2) On Twitter, users can navigate content
using hashtags, so we will use them to quantify the ease of moving across sub-
topics. Moreover, we point out that previous work focusing on change of topics
also exploited hashtags (VanDam and Tan, 2016).

Since 2018, more than 396K unique hashtags (excluding those consisting of a
single character) have appeared in $S_{\texttt{\\#MyBodyMyChoice}}$. To find
the most relevant hashtags in the stream, we sample 100K tweets from the
reference windows BP and the union of IP and CC111Because the volume of the
stream differs across reference windows, we sample with replacement to equally
weight the hashtags appearing in different periods. The size of the sample is
set to get a reasonable approximation of the real hashtags distribution.. We
take the union of the samples and extract a set $R_{i}$ of hashtags such that
at least 50% of the tweets in $S_{\texttt{\\#MyBodyMyChoice}}$ contains at
least one hashtag in $R_{i}$. We repeat the sampling 100 times and define the
set of hashtags $R=\cup_{i=1}^{100}R_{i}$ composed of 141 hashtags covering
57.39% of the entire stream of tweets.

The list $R$ can be grouped into the set $\mathcal{S}$ of five main sub-
topics: _women rights & abortion_, _politics & civil rights_, _freedom of
choice_ , _covid_ , and _vaccination_. We note that, for the periods after the
pandemic started, we group the _covid_ and _vaccination_ sub-topics together,
since we observed that (1) the two appear often together, and (2) there is no
other vaccine debate open during the time in analysis. We soft-assign222A
hashtag can belong to multiple topics. the hashtags in the list $R$ to each of
these sub-topics. We refer to the set $\mathcal{T}_{i}$ of hashtags assigned
to a sub-topic $i$ as its _signature_. In Table 1, we list some of the
hashtags contained in the signature of each sub-topic.

Sub-topic | Hashtags
---|---
_Women rights & abortion_ |  #prochoice, #womensrights,
#abortion, #abortionishealthcare,
#prolife, #abortionrights
_Politics & civil rights_ |  #blm, #blacklivesmatter,
#lgbtq, #votethemout
_Freedom of choice_ |  #freedom, #medicalfreedom, #righttochoose,
#freedomofchoice, #bodilyautonomy
_Covid_ |  #covid, #covid19, #nomorelockdowns,
#freedomofchoice, #bodilyautonomy
_Vaccination_ |  #vaccine, #novaccinepassports, #vaccinesideeffects,
#vaccinemandates, #iwillnotcomply

Table 1. For each sub-topic, we list some of the most relevant hashtags
contained in their signature.

Assign tweets to sub-topics. The process of assigning tweets to sub-topics
consists of two steps. In the first unsupervised step, given a tweet $t$, we
denote by $R_{t}$ the list of its hashtags that lie in $R$ and assign $t$ to
the sub-topic $i$ via

(1) $i=\underset{j\in\mathcal{S}}{\operatorname{argmax}}\ {\sum}_{h\in
R_{t}}\mathbb{P}\left(\mathcal{T}_{j}|h\right)\enspace.$

We apply this rule to all tweets that have at least one hashtag in $R$.
Unfortunately, some tweets may not have any hashtags in $R$. To address this
issue, we use the tweets labeled using Equation 1 as the ground truth to train
a supervised classifier on the text of the remaining set of tweets to assign a
tweet to a corresponding sub-topic. Our classifier consists of a Random Forest
(Breiman, 2001) of depth 35 for each time period, which is fed with the text
of tweets represented as a count vector333We evaluate each classifier on
5-folds (70/30) whose F-1 average is 83.75% (BP), 87.40% (IP), and 89.13%
(CC).. The average test F1-score on the sub-topics is 82% for the BP period,
84% for the IP period, and 90% for the CC period. We decided not to use
_covid-twitter-bert_ (Müller et al., 2020) since a Random Forest classifier is
an interpretable and simple model that already yields good results.
Interpretability of our model is important since our ground truth is
unsupervised, which provides extra flexibility if the labeled data we use
require additional cleaning.

##  3\. Analysis

We want to understand whether and how the discourse around the hashtag
#MyBodyMyChoice has changed after the Covid-19 outbreak by answering the
following questions.

RQ1. How has the usage of #MyBodyMyChoice evolved?

RQ2. How do user bases and content interactions differ across sub-topics?

RQ3. Has #MyBodyMyChoice been hashjacked?

###  3.1. RQ1: #MyBodyMyChoice Evolution

Hashtags function as means of labeling, archiving and distributing information
on OSNs. As such, we should not underestimate the importance of associating a
topic to a hashtag. #MyBodyMyChoice was originally created to group content
related to the _abortion_ debate. Thus, an average consumer of information on
Twitter would likely expect to find news and opinions about this topic when
searching for content associated with the hashtag.

In our attempt to answer RQ1, we investigate if a consumer’s experience is
likely to be consistent with this expectation, or whether the consumer is more
likely to encounter tweets related to the online protests against Covid-19
vaccine mandates in which the slogan of “My Body My Choice” is widely used to
claim their bodily autonomy.

#MyBodyMyChoice overview. For an overview of the discourse around
#MyBodyMyChoice, we analyse the hashtags appearing in our three reference
windows. In Figure 2, we observe that the hashtags used during the BP period
relate more frequently to the discourse centered around women’s rights, and in
particular abortion and body autonomy. During the IP period, women’s rights
still appear central in the discussion, with a specific focus on asking for
justice for women who were subjected to violence, e.g., #justiceforjennifer,
#justiceforuwa, and #merajismmerimarzi; at the same time, the hashtag #covid19
starts to gain relevance. During the CC period, corresponding to the first
vaccine approval in the U.S., we see that the conversation about
#MyBodyMyChoice is no longer centered around a topic associated to women’s
rights: instead, and in contrast to the periods preceding the Covid-19
pandemic, multiple hashtags related to the vaccination campaign become more
dominant.

Change of most relevant hashtags. We sample 100K tweets with replacement for
each period BP, IP, and CC. Given one of these periods $T$, let $H$ be the set
of the 25 hashtags used most often within the period $T$ and build the vector
$\textbf{v}_{H}^{T}\in\mathbb{R}^{25}$ whose entries are co-occurrences of
#MyBodyMyChoice with each of the 25 hashtags in $H$. In the time period $E$
following $T$, we count the co-occurrences of #MyBodyMyChoice with the same
set $H$ of hashtags, obtaining $\textbf{v}_{H}^{E}\in\mathbb{R}^{25}$. To
understand whether the frequency of co-occurrences of #MyBodyMyChoice with the
hashtags is independent of the time window, we run a $\chi^{2}$ test with a
significance threshold of $5\%$. Applying this framework to the periods $T=BP$
and $E=IP$, we observe that the frequency of the hashtags is not independent
of the time window. For the periods $T=IP$ and $E=CC$, we find similarly that
the frequency of hashtags is not independent of the time window according to
the $\chi^{2}$ test.

Observation 1. The Covid-19 outbreak caused a decrease of occurrences of the
25 hashtags used most often before the pandemic. Similarly, the beginning of
the vaccination campaign caused a change of occurrences of the hashtags that
were most dominant during IC. For instance, the frequency of the hashtags
#abortion, #prochoice, and #merajismmerimarzi was reduced by the 48.66%,
52.82%, and 98.35% respectively, while the hashtags #covid19, #vaccine, and
#freedom increased in frequency by 97.73%, 422.66%, and 690.82% respectively.

Sub-topics monopolization. Following the process explained in Section 2.1, we
separate the tweets into sub-topics. We refer to the fraction of tweets
assigned to a sub-topic as the sub-topic’s monopolization factor. We note
that, for the periods after the pandemic started, we grouped together _covid_
and _vaccination_ sub-topics, since we observed that (1) the two appear often
together, and (2) there is no other vaccine debate open during the time in
analysis. Figure 3 shows the monthly monopolization factor of each of the sub-
topics. Consistently with our observations of the most dominant hashtags shown
in Figure 2, the fraction of tweets about _women’s right & abortion_ starts to
decrease after the first official Covid-19 case was registered outside China,
and the start of the vaccination campaign further reduces the original sub-
topic of the hashtag #MyBodyMyChoice with the introduction of hashtags such as
#novaccinepassport, #novaccinemandate, and #dontcomply. In fact, during the
second half of 2021, the tweets about _women’s right & abortion_ represent
less than 40% of the total number of tweets. Note, however, that the
monopolization factor of _covid_ is reduced after September 2021 when the
abortion ban in Texas went into effect.

Observation 2. After the Covid-19 outbreak, the usage of the hashtag
#MyBodyMyChoice changed on Twitter, with the conversation drifting towards
Covid-19. In particular, the most relevant hashtags after the outbreak reflect
opinions opposing the current vaccination campaign, typical of antivaxxers.

Figure 3. Sub-Topic Monopolization: for each sub-topic, we show the monthly
fraction of tweets belonging to it.

Sub-topics separation. So far, we have investigated how the original topic of
_abortion_ in the #MyBodyMyChoice stream was affected by the Covid-19
breakout. Next, we want to understand how connected or separated the four sub-
topics are from each other. To this end, we built a hashtag network, where
each node is a hashtag, and two hashtags are connected by an edge if they co-
occur in at least 30 tweets. Each edge is weighted by the number of co-
occurrences. Figure 4 shows a planar embedding of the resulting network in
which nodes are colored by sub-topic. The figure indicates that each sub-topic
corresponds to a separate set of hashtags. To quantify this phenomenon, we
employ the Random-Walk Controversy score (RWC) (Garimella et al., 2018). The
RWC score measures the degree of separation between two sets of nodes in a
graph by comparing the probabilities of moving within and between the two
sets. A negative score indicates that the two sets of nodes are not separated
well so that the number of connections within and across the sets are
comparable. When the RWC score is larger than zero, the two sets are well-
separated, and there are likely very few connections between them relative to
the number of connections within them. In our framework, the RWC score
quantifies the overall separation of sub-topics by measuring the frequency
with which hashtags associated with different sub-topics co-occur. On Twitter,
hashtags are hyperlinks that allow users to navigate the associated content.
Thus, in this context, RWC scores reflect how much users who navigate the
content of a sub-topic are exposed to links related to content about other
sub-topics. In particular, a lower RWC score implies that users who explore a
sub-topic are less likely to reach content about the other sub-topic if they
are navigating Twitter by following hashtags.

Figure 4. Network of hashtags in the #MyBodyMyChoice stream. We draw the
largest connected components of the network. Pink nodes refer to the _women
rights & abortion_ sub-topic, green nodes to _covid_ and _vaccination_ , blue
nodes to _politics & civil rights_, and the orange nodes to _freedom of
choice_.

Sub-topic | _Women’s rights & abortion_ | _Politics & civil rights_ | _Freedom of choice_ | _Covid & Vaccination_
---|---|---|---|---
_Women rights & abortion_ | - | 0.49 | 0.32 | 0.46
_Politics & civil rights_ | 0.49 | - | 0.31 | 0.48
_Freedom of choice_ | 0.32 | 0.31 | - | 0.11
_Covid & Vaccination_ | 0.46 | 0.48 | 0.11 | -

Table 2. For each pair of sub-topics, we report the RWC score. The closer the
score is to one, the more separated the two sub-topics are. Note that the RWC
score is symmetric.

Table 2 reports the RWC scores for each pair of sub-topics. The _Women’s
rights & abortion_ and the _covid & vaccination_ sub-topics are the most
separated, thus illustrating the topics shift of #MyBodyMyChoice. In contrast,
_covid & vaccination_ is close to _freedom of choice_ , which, in turn, is
close to _politics & civil rights_.

Observation 3. The _women’s rights & abortion_ and _covid & vaccination_ sub-
topics are well-separated. In particular, we expect that the hashtags
characterizing the first sub-topic rarely co-occur with the hashtags
representing the second sub-topic. Users who access content related to one of
the two sub-topics therefore have a smaller probability of seeing hyperlinks
to content related to the other sub-topic than to content of the same sub-
topic.

Figure 5. A list of the ten most retweeted users within the _women’s rights &
abortion_ and the _covid & vaccination_ sub-topics. Percentages are computed
with respect to the total number of retweets.

###  3.2. RQ2: Compare Sub-Topics

The analysis in the previous section showed that Covid-19 has become a
prominent sub-topic of the stream of tweets tied to #MyBodyMyChoice. In this
section, our goal is to understand how the sub-topics related to Covid-19 and
abortion differ from each other in terms of their user bases and the
popularity of content. We consider the tweets posted during the IP and CC
periods, because the _covid & vaccination_ sub-topic pop up after the
beginning of the pandemic. The number of unique users that posted content
about _women’s rights & abortion_ is 1.82 times the number of users tweeting
about _Covid-19 & vaccines_. Furthermore, 90.7% of users tweeted about only
one of the two topics. Thus, we conclude that the user base of the two
branches of #MyBodyMyChoice are separated.

Next, we investigate whether the number of posts, retweets, replies, and
quotes, which constitute the different ways in which tweets are generated,
differ between these two sub-topics and find that there are no significant
differences: For the sub-topic _women’s rights & abortion_, 56.71% of tweets
are retweets, 18.0% are replies, 7.34% are quotes, and the remaining are
posts; the tweets that generate the retweets constitute 11.85% of the total
tweets in this sub-topic. In the corpus of _covid & vaccination_, 55.94% of
tweets are retweets, 20.92% are replies, 6.38% are quotes, and the remaining
are posts; the tweets that generate the retweets make up 12.51% of all tweets
in this sub-topic. Finally, we observe that the distributions of the number of
tweets per user is comparable for both sub-topics: the median is one with few
outliers in both populations. We speculate that the similar distribution
across the sub-topics indicates that users tweeting about _covid &
vaccination_ were not part of an organized effort to intentionally shift the
use of the hashtag #MyBodyMyChoice.

Observation 4. The user bases of _women’s rights & abortion_ and _covid &
vaccination_ are different and display a similar characterization of content
interaction (i.e., posts, retweets, replies, and quotes are distributed
similarly). This indicates that there was no organized intervention to
intentionally create a shift of the #MyBodyMyChoice discourse.

Figure 6. Listed are the domains of the most frequent URLs attached to tweets
for the _women’s rights & abortion_ (top) and _covid & vaccination_ (bottom)
sub-topics. The percentages are computed over the total number of tweets that
contain a non-Twitter URL. The percentage of tweets containing non-Twitter
URLs is 33.1% for _women’s rights & abortion_ and 25.08% for _covid &
vaccination_.

In Figure 5, we report the top-10 retweeted users appearing in tweets about
_women’s rights & abortion_ and _covid & vaccination_, respectively. The group
of most frequently retweeted users focused on the topic of _women’s rights &
abortion_ are divided fairly evenly between a few categories of users. These
include activist and reproductive rights organizations, individuals from film,
medical, and academic circles, and users known primarily for their social
media presence. In contrast, the group of most retweeted users focused on the
topic of _covid & vaccinations_ includes more users from the social media
category, fewer authority figures, and no organizations. In the latter sub-
topic, there is also a striking gap between the most influential user’s share
of retweets and everyone else’s: the user @GillianMcKeith has 37% of the
retweets in this category, while the rest of the top ten have 1-3% each.
Furthermore, three of the top ten accounts in the second sub-topic have been
suspended.

Observation 5. The majority of the most retweeted users of _women’s rights &
abortion_ are organizations, activists, or professionals. In contrast, the set
of accounts retweeted by the user base of _covid & vaccination_ suggests that
it gives more prominence to users that individually select, create, and share
content without the support of established organizations.

In Figure 6, we report the URLs appearing in tweets from the _women’s rights &
abortion_ and _covid & vaccinations_ sub-topics. The domains linked to in
tweets from the _women’s rights & abortion_ subcategory come primarily from
three categories of sources; social media, traditional news outlets, and
websites advocating for a position or providing information. The group of top
domains linked to in tweets from the _covid & vaccinations_ subcategory adds
four additional social media sites to the four used by the abortion
subcategory. These additional sites are known for having less restrictive
censorship policies. Advocacy groups have a larger share of the links amongst
the _covid and vaccination_ subgroup, and they are a different set of
organizations than those linked to by the women’s rights and abortion
subcategory. Finally, traditional news outlets make up a much smaller portion
of the top domains in the _covid and vaccinations_ subcategory, and again the
specific outlets are very different. In addition, one outlet known to
disseminate ‘fake news’ is amongst the top twenty.

###  3.3. RQ3: Has #MyBodyMyChoice been hijacked?

There are a variety of different types of hashjacking, and this section
situates the case of #MyBodyMyChoice in the context of other well-known
hashjackings based on the analytical observations drawn so far. There are
largely three types of hashjackings: those that semantically change the
associated tweets, those that intentionally make a topic more polarized, and
those that capitalize on the success of a given hashtag without taking part in
the original conversation. These three categories are outlined below:

1) Hashjacking that semantically changes the associated tweets (whether
intentional or not). When Alyssa Milano, in 2017, revitalized the #Metoo
movement on Twitter by encouraging thousands of women to share their stories
of sexual harassment, the movement garnered widespread global attention that
did not attribute its founder, Tarana Burke, or use the language of the
movement within its intended context of the Black community. As a result,
#Metoo turned from a movement focused on Burke’s community to women’s
experiences with sexual harassment broadly (Times, 2018). While the women
responding to Milano’s encouragement to post their own stories were not
necessarily intentionally hashjacking Burke’s words for the Black community,
the result was that the semantic meaning of the hashtag and its associations
changed.

2) Hashjacking that intentionally makes a topic more polarized. Within this
category of hashjackings, users appropriate a hashtag for the opposite purpose
of the hashtag’s originally intended use. For example, McDonald’s launched a
Twitter campaign using the hashtag #McDStories to encourage users to share
their fond stories of Happy Meals. The hashtag took off as users posted
critical comments about the food and how it impacted their health negatively.
A similar example is the hashtag #Februdairy, which was launched by the diary
industry to support the dairy market after the creation of Veganuary, a
challenge to join a global community to try vegan food for a month: users
repurposed the hashtag to write about problems in the dairy industry.

3) Hashjacking to capitalize on the success of a given hashtag without taking
part in the conversation. After #Metoo was popularized by Alyssa Milano, the
hashtag was then also hashjacked in 2018 by the ’120 decibels’ campaign by the
Austrian Identitarian Movement in an attempt to latch onto to prominence of
the #Metoo hashtag (Knüpfer et al., 2020). Similarly, the hashtag #Brazil2014
was hashjacked by ISIS in order to entice fans watching the World Cup in 2014
to click on links to the group’s propaganda videos (Farwell, 2014). In these
hashjackings, the hashjackers did not intend to engage with the topic of the
hashtag; rather they were solely using the hashtag to capitalize on its
success.

Observation 6. #MyBodyMyChoice was a hashtag rooted in women’s rights and was
then used by users during the pandemic to refer to something else entirely:
anti-vaccine content. We situate #MyBodyMyChoice within the semantic
hashjacking as (1) the discourse around the hashtag changed over time, and
users who were hashjacking are distributed unique users rather than a given
organization (Observations 1, 2, 4, & 5); (2) Users who hashjacked
#MyBodyMyChoice used it to summarize their own positions on vaccines and masks
rather than capitalize on the hashtag’s success. Furthermore, the hashtag was
not hashjacked to make women’s reproductive rights more polarized but rather
to express a different argument against COVID-19 vaccines (Observation 3).

##  4\. Related Work

We contextualize our work by providing a brief overview of previous studies
about hashjacking. Some of these works analyse whether a hashtag has changed
its usage (i.e., hashjacked), while some concentrate on developing tools to
identify semantic shifts.

Past works emphasized the change of usage of a hashtag due to an event or by
group of users. Rodriguez (2020) showed that #FamilyTravelHacks, created to
inform people about tips for safe travel with children, changed its usage
after Executive Order 13769, a.k.a., the travel ban. As described in Section
3.3, Knüpfer et al. (2020) demonstrated that the hashtag #metoo was hijacked
by the ‘120 decibels’ campaign. Some other works analyzed how polarized users
intentionally _hash-jack_ a tweet to express their views contrasting the
opinion of the original user base of the hashtag, e.g., in Germany’s
presidential elections (Darius and Stephany, 2019) or in U.S. politics (Hadgu
et al., 2013).

To analyse these phenomena, several methodological works focus on designing
computational tools to identify changes in the linguistic usage of a word
(Kulkarni et al., 2015; Danescu-Niculescu-Mizil et al., 2013; Hamilton et al.,
2016) or of the meaning of a hashtag on Twitter (VanDam and Tan, 2016).
Particularly, VanDam and Tan (2016) use a matrix factorization-based method to
build a day-by-day topic matrix and use a standard change-detection hypothesis
test to see if a change in the topic occurred. While they provide a strategy
to say “if” the topic of a hashtag has changed or not, we focus on “how” the
meaning of #MyBodyMyChoice has changed. For this reason, we adopted a
qualitative approach instead of their method.

##  5\. Conclusion

In this paper, we examined the hashtag #MyBodyMyChoice to see how the content
of tweets associated with it evolved during the pandemic. The hashtag
originated to support women’s rights around abortion and bodily autonomy, but
we observed that during the pandemic a separate set of users utilized it to
communicate their disagreement with vaccines.

We analysed the change of usage of #MyBodyMyChoice considering three factors:
(1) the semantic of the discourse around the hashtag; (2) the users’
interactions with contents related to diverse sub-topics; and (3) the users’
attention toward specific users and external sources of information. We
demonstrate the hypothesized semantic shift by employing supervised learning
and statistical analysis. Furthermore, we quantify the shift by showing the
evolution of dominant sub-topics. To understand whether the semantic shift was
intentionally induced, we conducted a user behaviour analysis and found no
evidence for such organized attempts. Finally our analysis of users’ attention
to highly retweeted users and most commonly embedded URLs shows that attention
has shifted from experts and authorities to individuals and that some
unreliable sources appeared during the Covid-19 debate. We note that our
analysis focused only on the stream of tweets using #MyBodyMyChoice and does
therefore not necessarily generalize to the entire discourse of these topics.
Moreover, although the methodology we used can be applied to other topics, it
would still require human in the loop to first distinguish among sub-topics.

Disclaimer. It is important to note that the demographics of Twitter users is
not representative of the overall population: 40% of Twitter users are in the
25-34 age group, another 20% are in the 35-49 age group, and the age groups
18-24 and above 50 each constitute around 17% of Twitter users (Statista);
furthermore, 63% of Twitter users are male. Thus, as our study explores how
the content associated with the #MyBodyMyChoice hashtag changed over the
pandemic, it is important to keep in mind that our results may have been
impacted by the demographics on Twitter.

##  6\. Acknowledgments

This work was a result of collaboration between Brown University’s Data
Science Initiative, Brown University’s Computer Science Department and the
Center for Digital Scholarship at Brown University. Haddadan was partially
supported by the Data Science Initiative at Brown University and by the NSF
via grant CCF-1740741. Ramachandran acknowledges support from US National
Institutes of Health R01 GM118652 and R35 GM139628. Sandstede acknowledges
support from the NSF grant CCF-1740741.

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# Sharp and quantitative estimates for the $p-$Torsion of convex sets

Amato V., Masiello A. L., Paoli G., Sannipoli R

###### Abstract

Let $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$, be a bounded, open and convex
set and let $f$ be a positive and non-increasing function depending only on
the distance from the boundary of $\Omega$. We consider the $p-$torsional
rigidity associated to $\Omega$ for the Poisson problem with Dirichlet
boundary conditions, denoted by $T_{f,p}(\Omega)$. Firstly, we prove a Pólya
type lower bound for $T_{f,p}(\Omega)$ in any dimension; then, we consider the
planar case and we provide two quantitative estimates in the case $f\equiv 1$.

MSC 2020: 35P15, 35J05, 35J25, 49Q10, 47J10.
Keywords: $p-$Laplacian, $p-$Torsional rigidity, Pólya estimates, quantitative
inequalities.

##  1 Introduction

Let $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$, be a bounded, open and convex
set with non-empty interior and let $p\in(1,+\infty)$. We consider the Poisson
equation for the $p-$Laplace operator, defined as

$-\Delta_{p}u:=-{\rm div}\left(|\nabla u|^{p-2}\nabla u\right),$

with Dirichlet boundary condition:

$\begin{cases}-\Delta_{p}u(x)=f(d(x,\partial\Omega))&\text{ in }\Omega\\\
u=0&\text{ on }\partial\Omega,\end{cases}$ (1)

where $f:[0,R_{\Omega}]\to[0,+\infty[$ is a continuous, non-increasing and not
identically zero function, $d(\cdot,\partial\Omega):\Omega\to[0,+\infty[$ is
the distance function from the boundary defined as

$d(x,\partial\Omega):=\inf_{y\in\partial\Omega}{\left|x-y\right|}$

and $R_{\Omega}$ is the inradius of $\Omega$, i.e.

$R_{\Omega}=\sup_{x\in\Omega}d(x,\partial\Omega).$

This class of functions, depending only on the distance, are the so called web
functions, see as a reference [8]. A function $u\in W^{1,p}_{0}(\Omega)$ is a
weak solution to (1) if and only if

$\int_{\Omega}|\nabla u(x)|^{p-2}\nabla
u(x)\nabla\varphi(x)\,dx=\int_{\Omega}f(d(x,\partial\Omega))\varphi(x)\,dx\quad\forall\varphi\in
W^{1,p}_{0}(\Omega).$

The $(f,p)$-torsional rigidity of $\Omega$, that we denote by
$T_{f,p}(\Omega)$, is defined as

$T_{f,p}(\Omega)=\max_{\begin{subarray}{c}\varphi\in W^{1,p}_{0}(\Omega)\\\
\varphi\not\equiv
0\end{subarray}}\frac{\displaystyle{\left(\int_{\Omega}f(d(x,\partial\Omega)){\left|\varphi(x)\right|}\,dx\right)^{\frac{p}{p-1}}}}{\left(\displaystyle{\int_{\Omega}{\left|\nabla\varphi(x)\right|}^{p}\,dx}\right)^{\frac{1}{p-1}}}$
(2)

and, if $u_{p}\in W^{1,p}_{0}(\Omega)$ is the unique solution to (1), we have

$T_{f,p}(\Omega)=\int_{\Omega}fu_{p}\,dx.$

For the sake of simplicity, when $f\equiv 1$ in $\Omega$, we set
$T_{p}(\Omega):=T_{1,p}(\Omega)$ and, if we are also in the case $p=2$, we set
$T(\Omega):=T_{1,2}(\Omega)$.

In what follows, we denote by $|\Omega|$ and $P(\Omega)$ respectively the
Lebesgue measure and the perimeter of $\Omega$ in the sense of De Giorgi. In
[18] the author gives some estimates on the torsional rigidity $T(\Omega)$. In
particular, he proves that, among all bounded, open and convex planar sets,
the following inequality holds

$\frac{T(\Omega)P^{2}(\Omega)}{{\left|\Omega\right|}^{3}}\geq\frac{1}{3}$ (3)

and equality is asymptotically achieved by a sequence of thinning rectangles.
Moreover, Makai in [16] proves that among all bounded, open and convex planar
sets, the following upper bound holds

$\frac{T(\Omega)P^{2}(\Omega)}{{\left|\Omega\right|}^{3}}\leq\frac{2}{3},$ (4)

which is sharp on a sequence of thinning triangles (for the exact definition
of thinning domains see Definition 9). Estimates (3) and (4) are generalized
to the $p-$Laplacian in [10]. More precisely, the authors prove that, in the
class of bounded, open and convex planar sets,

$\frac{1}{q+1}<\frac{T_{p}(\Omega)P^{q}(\Omega)}{{\left|\Omega\right|}^{q+1}}<\frac{2^{q+1}}{(q+2)(q+1)}\qquad
q=\frac{p}{p-1},$ (5)

where the lower and the upper bounds hold asymptotically on a sequence of
thinning rectangles and on a sequence of thinning isosceles triangles,
respectively. In [9] the authors generalize the lower bound (5) in every
dimensions, proving that, for bounded, open and convex sets
$\Omega\subseteq\mathbb{R}^{n}$, it holds

$\frac{T_{p}(\Omega)P^{q}(\Omega)}{{\left|\Omega\right|}^{q+1}}>\frac{1}{q+1},$
(6)

and they extend such result also to the anisotropic case.

We also recall that in [7] the authors consider the functional

$H_{k}(\Omega)=\dfrac{P(\Omega)T^{k}(\Omega)}{|\Omega|^{\alpha_{k}}}\qquad\alpha_{k}=1+k+\frac{2k-1}{n}$

and prove that, among bounded, open and convex sets in $\mathbb{R}^{n}$, this
functional is bounded if and only if $k=1/2$. More precisely, they prove the
following:

$\frac{1}{\sqrt{3}}\leq
H_{\frac{1}{2}}(\Omega)\leq\frac{2^{n}n^{3n/2}}{\omega_{n}}\left(\frac{n}{n+2}\right)^{\frac{1}{2}},$
(7)

where $\omega_{n}$ is the Lebesgue measure of the unit ball. We note that, in
the planar case, the lower bound in (7) coincides with the one given in (3),
while the upper bound is strictly larger than the one given in (4). It is
conjectured that, in the higher dimensional case, the upper bound is

$H_{\frac{1}{2}}(\Omega)\leq
n\left(\frac{2}{(n+1)(n+2)}\right)^{\frac{1}{2}}.$

Moreover, we observe that the lower bound in (7) is asymptotically achieved by
a sequence of thinning cylinders. More precisely, denoting by $w_{\Omega}$ the
minimal width and by $\operatorname{\,\textup{diam}}(\Omega)$ the diameter of
the set (see Section 2 for the exact definitions), we give the following

######  Definition 1.1.

Let $\Omega_{l}$ be a sequence of bounded, open and convex sets of
$\mathbb{R}^{n}$. We say that $\Omega_{l}$ is a sequence of thinning domains
if

$\dfrac{w_{\Omega_{l}}}{\operatorname{\,\textup{diam}}(\Omega_{l})}\xrightarrow{l\to
0}0.$ (8)

$\operatorname{\,\textup{diam}}(\Omega_{l})$$w_{\Omega_{l}}$$\Omega_{l}$

In particular, if $l>0$ and $C$ is a bounded, open and convex set of
$\operatorname{\mathbb{R}}^{n-1}$ with unitary $(n-1)$-dimensional measure,
then, if $l\to 0$, the sequence

$\Omega_{l}=l^{-\frac{1}{n-1}}C\times\left[-\frac{l}{2},\frac{l}{2}\right]$
(9)

is called a sequence of thinning cylinders. Moreover, in the case $n=2$, the
sequence (9) is called sequence of thinning rectangles.

$l^{-\frac{1}{n-1}}C$$l$$\Omega_{l}$

The first result that we prove, following the method of proof used in [18]
with the use of web functions as test functions, is a lower bound for the
$(f,p)$-torsional rigidity, which generalizes the lower bound in (5).

######  Theorem 1.1.

Let $\Omega$ be a bounded, open and convex set with non-empty interior of
$\operatorname{\mathbb{R}}^{n}$, $n\geq 2$, and let
$f:[0,R_{\Omega}]\to[0,+\infty[$ be a continuous and non-increasing function
such that $f\not\equiv 0$. Then, it holds

$T_{f,p}(\Omega)\geq c_{p}\frac{\mu^{q+1}_{f}(\Omega)}{f(0)P^{q}(\Omega)},$
(10)

where

$c_{p}=\frac{p-1}{2p-1},\qquad q=\frac{p}{p-1},$

and

$\mu_{f}(\Omega)=\int_{\Omega}f(x)\,dx.$

Moreover, the equality sign is asymptotically achieved by a sequence of
thinning cylinders.

We stress that both the estimate and the constant in Theorem 1.1 are
independent of the dimension of the space.

In the second part of the present paper, we focus our study on the case
$f\equiv 1$ and $n=2$ and we obtain some quantitative estimates. We define the
following functional

$\mathcal{F}_{p}(\Omega)=\frac{T_{p}(\Omega)P^{q}(\Omega)}{|\Omega|^{q+1}}\qquad
q=\frac{p}{p-1},$ (11)

which is scaling invariant, since for every $t>0$

${\left|t\Omega\right|}=t^{n}{\left|\Omega\right|},\qquad
P(t\Omega)=t^{n-1}P(\Omega)$

and

$T_{p}(t\Omega)=t^{n+q}T_{p}(\Omega).$

We can rewrite inequality (10), in the case $f\equiv 1$, as follows

$\mathcal{F}_{p}(\Omega)\geq c_{p}.$

From Theorem 1.1 follows that along a sequence of thinning cylinders
$\\{\Omega_{l}\\}_{l\in\mathbb{N}}$ defined in (9), we have

$\mathcal{F}_{p}(\Omega_{l})\xrightarrow{l\to 0}c_{p}.$

This leads to the following stability issue: if $\mathcal{F}_{p}(\Omega)$ is
close to $c_{p}$, can we say that $\Omega$ is close in some sense to a
cylinder? The following result gives us information on the nature of the
geometry of $\Omega$: when $\mathcal{F}_{p}(\Omega)-c_{p}$ is sufficiently
small, the set $\Omega$ is a thin domain, in the sense that the ratio
$w_{\Omega}/{\rm diam}(\Omega)$ is small.

The main novelty of the present paper consists indeed in the following
quantitative results of the Polya estimates (3) and the Polya type lower bound
in (5) by means of suitable deficits. For completeness, we recall some
standard references about isoperimetric quantitative results, see for example
[11, 12, 6, 5, 13, 17]. The main difference between these results and ours is
that the equality in Polya’s estimates is achieved asymptotically for a
sequence of thinning cylinders. Hence, the proof of quantitative result must
take into account that we do not have a minimum, as in the classical
isoperimetric stability results.

######  Theorem 1.2.

Let $\Omega$ be a bounded, open and convex set with non-empty interior of
$\operatorname{\mathbb{R}}^{2}$ and let $f\equiv 1$. Then,

$\mathcal{F}_{p}(\Omega)-c_{p}\geq
K(p)\frac{w_{\Omega}}{\operatorname{\,\textup{diam}}(\Omega)},$ (12)

where $K(p)$ is a positive constant that can be computed explicitly:

$K(p)=\frac{(p-1)p}{2^{\frac{p}{p-1}}3(3p-2)(2p-1)}.$ (13)

Moreover, the exponent of the quantity
$\displaystyle\frac{w_{\Omega}}{\operatorname{\,\textup{diam}}(\Omega)}$ is
sharp.

We prove a second quantitative result in the case $p=2$.

######  Theorem 1.3.

Let $\Omega$ be a bounded, open and convex set with non-empty interior in
$\operatorname{\mathbb{R}}^{2}$, let $f\equiv 1$ and let $p=2$. Then, there
exists a positive constant $\tilde{K}$ such that

$\mathcal{F}_{2}(\Omega)-c_{2}=\frac{T(\Omega)P^{2}(\Omega)}{|\Omega|^{3}}-\frac{1}{3}\geq\tilde{K}\left(\frac{|\Omega\bigtriangleup
Q|}{|\Omega|}\right)^{3},$ (14)

where $\Omega\bigtriangleup Q$ denotes the symmetric difference between
$\Omega$ and a rectangle $Q$ with sides $P(\Omega)/2$ and $w_{\Omega}$
containing $\Omega$.

We prove the first quantitative result (12) starting from the proof of Theorem
1.1, estimating the remainder term with geometric quantities. Moreover, we
partially extend this last result in every dimension (see Remark 4.1).
Finally, in order to prove (14), we use Steiner formulas (see Section 2.2 and
the references therein).

The paper is organized as follows. In Section $2$ we recall some basic
notions, definitions and we recall some classical results, focusing in
particular on the class of convex sets. In Section $3$ we prove Theorem 1.1
and, finally, Section $4$ is dedicated to the proof of the quantitative
results in the planar case, when $f\equiv 1$.

##  2 Notations and Preliminaries

###  2.1 Notations and basic facts

Throughout this article, $|\cdot|$ will denote the Euclidean norm in
$\mathbb{R}^{n}$, while $\cdot$ is the standard Euclidean scalar product for
$n\geq 2$. By $\mathcal{H}^{k}(\cdot)$, for $k\in[0,n)$, we denote the
$k-$dimensional Hausdorff measure in $\mathbb{R}^{n}$.

The perimeter of $\Omega$ in $\mathbb{R}^{n}$ will be denoted by $P(\Omega)$
and, if $P(\Omega)<\infty$, we say that $\Omega$ is a set of finite perimeter.
In our case, $\Omega$ is a bounded, open and convex set; this ensures us that
$\Omega$ is a set of finite perimeter and that
$P(\Omega)=\mathcal{H}^{n-1}(\partial\Omega)$. Moreover, if $\Omega$ is an
open set with Lipschitz boundary, it holds

######  Theorem 2.1 (Coarea formula).

Let $f:\Omega\to\operatorname{\mathbb{R}}$ be a Lipschitz function and let
$u:\Omega\to\operatorname{\mathbb{R}}$ be a measurable function. Then,

${\displaystyle\int_{\Omega}u|\nabla
f(x)|dx=\int_{\mathbb{R}}dt\int_{(\Omega\cap
f^{-1}(t))}u(y)\,d\mathcal{H}^{n-1}(y)}.$ (15)

Some references for results relative to the sets of finite perimeter and for
the coarea formula are, for instance, [15, 1].

Recall that the inradius $R_{\Omega}$ of $\Omega$ is

$R_{\Omega}=\sup_{x\in\Omega}\inf_{y\in\partial\Omega}|x-y|.$ (16)

We give now the definition of the support function of a convex set and minimal
width (or thickness) of a convex set.

######  Definition 2.1.

Let $\Omega$ be a bounded, open and convex set of $\mathbb{R}^{n}$. The
support function of $\Omega$ is defined as

$h_{\Omega}(y)=\max_{x\in\Omega}\left(x\cdot y\right),\qquad
y\in\mathbb{R}^{n}.$

######  Definition 2.2.

Let $\Omega$ a bounded, open and convex set of $\mathbb{R}^{n}$, the width of
$\Omega$ in the direction $y\in\mathbb{R}$ is defined as

$\omega_{\Omega}(y)=h_{\Omega}(y)+h_{\Omega}(-y)$

and the minimal width of $\Omega$ as

$w_{\Omega}=\min\\{\omega_{\Omega}(y)\,|\;y\in\mathbb{S}^{n-1}\\}.$

We recall the following estimate, which is proved in [3] in the planar case
and is generalized in [4] in every dimensions.

######  Proposition 2.2.

Let $\Omega$ be a bounded, open and convex set of $\mathbb{R}^{n}$ with non-
empty interior. Then,

$\dfrac{1}{n}\leq\dfrac{|\Omega|}{P(\Omega)R_{\Omega}}<1.$ (17)

The upper bound is sharp on a sequence of thinning cylinders, while the lower
bound is sharp, for example, on balls. Moreover, for $n=2$, any circumscribed
polygon, that is a polygon whose incircle touches all the sides, verifies the
lower bound with the equality sign.

In the planar case the following inequalities hold true (see as a reference
[22, 21, 19]).

######  Proposition 2.3.

Let $\Omega$ be a bounded, open and convex set of $\mathbb{R}^{2}$. Then,

$2\leq\frac{w_{\Omega}}{R_{\Omega}}\leq 3.$ (18)

The upper bound is achieved by equilateral triangles and the lower bound is
achieved by disks.
Moreover,

$\left(w_{\Omega}-2R_{\Omega}\right)P(\Omega)\leq\dfrac{2}{\sqrt{3}}w^{2}_{\Omega},$
(19)

with equality holding for equilateral triangles, and

$|\Omega|\leq R_{\Omega}\left(P(\Omega)-\pi R_{\Omega}\right),$ (20)

with equality holding for the stadii (convex hull of two identical disjoint
balls).
Eventually,

$2\operatorname{\,\textup{diam}}(\Omega)<P(\Omega)\leq\pi\operatorname{\,\textup{diam}}(\Omega),$
(21)

where the lower bound is asymptotically achieved by a sequence of thinning
rectangles and the upper bound by sets of constant width.

###  2.2 Inner parallel sets

Let $\Omega$ be a bounded, open and convex set of
$\operatorname{\mathbb{R}}^{n}$ with non empty interior. The distance function
from the boundary is defined as

$d(x,\partial\Omega)=\inf_{y\in\partial\Omega}{\left|x-y\right|},$

and we will denote it by $d(\cdot)$. We remark that the distance function is
concave, as a consequence of the convexity of $\Omega$.

The superlevel sets of the distance function

$\Omega_{t}=\set{x\in\Omega\,:\,d(x)>t},\qquad t\in[0,R_{\Omega}]$ (22)

are called _inner parallel sets_ , where we denote by $R_{\Omega}$ the
inradius (see for the definition(16)), and we use the following notations:

$\mu(t)={\left|\Omega_{t}\right|},\qquad P(t)=P(\Omega_{t})\qquad
t\in[0,R_{\Omega}].$ (23)

By coarea formula (15), recalling that ${\left|\nabla d\right|}=1$ almost
everywhere, we have

$\mu(t)=\int_{\Set{d>t}}\,dx=\int_{\Set{d>t}}\frac{{\left|\nabla
d\right|}}{{\left|\nabla
d\right|}}\,dx=\int_{t}^{R_{\Omega}}\frac{1}{{\left|\nabla
d\right|}}\int_{\Set{d=s}}d\mathcal{H}^{n-1}\,ds=\int_{t}^{R_{\Omega}}P(s)\;ds;$

hence, the function $\mu(t)$ is absolutely continuous, decreasing and its
derivative is $\mu^{\prime}(t)=-P(t)$ almost everywhere. Moreover, it is
possible to prove that the perimeter $P(t)$ is non increasing and absolutely
continuous, as a consequence of the concavity of the distance function and the
Brunn-Minkowski inequality for the perimeter ( see [20] as a reference).

Finally, let us consider the case $n=2$. For $\Omega$ a bounded, open and
convex set of $\mathbb{R}^{2}$ with nonempty interior the Steiner formulas for
the inner parallel sets hold (see [2]):

$P(t)\leq P(\Omega)-2\pi t\qquad\forall t\in[0,R_{\Omega}],$ (24)
$\mu(t)\geq|\Omega|-P(\Omega)t+\pi t^{2}\qquad\forall t\in[0,R_{\Omega}],$
(25)

equality holding in both (24) and (25) for the stadii (see [10]). From (24),
we have that, if $\Omega$ is a convex set, then

$-P^{\prime}(t)\geq 2\pi,$ (26)

with equality if $\Omega$ is a ball or a stadium.

##  3 Proof of Theorem 1.1

In this Section we prove Theorem 1.1. Since the proof is quite long, we split
it in two parts: firstly we prove inequality (10) and, then, we prove its
sharpness.

#### Step 1: proof of inequality (10) in Theorem 1.1

###### Proof.

Let us choose in the variational characterization (2) $\varphi(x)=g(d(x))$ as
a test function, where $g$ is a positive and non-decreasing function in
$W^{1,p}([0,R_{\Omega}])$ such that $g(0)=0$. Then, by coarea formula (15),

$\int_{\Omega}f(d(x,\partial\Omega))\varphi(x)\,dx=\int_{0}^{R_{\Omega}}f(t)g(t)P(t)\,dt$
(27)

and

$\int_{\Omega}{\left|\nabla\varphi(x)\right|}^{p}\,dx=\int_{0}^{R_{\Omega}}g^{\prime
p}(t)P(t)\,dt.$ (28)

By (2), (27) and (28) we have

$T_{f,p}(\Omega)\geq\frac{\displaystyle{\left(\int_{0}^{R_{\Omega}}f(t)g(t)P(t)\,dt\right)^{\frac{p}{p-1}}}}{\left(\displaystyle{\int_{0}^{R_{\Omega}}g^{\prime
p}(t)P(t)\,dt}\right)^{\frac{1}{p-1}}}.$ (29)

Now, if we define the following measure

$\mu_{f}(E)=\int_{E}f(d(x))\,dx,$

we have

$\mu_{f}(t):=\mu_{f}(\Omega_{t})=\int_{\Omega_{t}}f(d(x))\,dx=\int_{t}^{R_{\Omega}}f(s)P(s)\,ds$
(30)

and, since $f(s)P(s)$ is a decreasing function, we get

$\mu_{f}(t)\leq(R_{\Omega}-t)f(t)P(t).$ (31)

From (30), we have

$-\mu^{\prime}_{f}(t)=f(t)P(t)\qquad\text{ a.e. }t\in[0,R_{\Omega}].$ (32)

Using (27), (32) and integrating by parts, we obtain

$\int_{0}^{R_{\Omega}}f(t)g(t)P(t)\,dt=-\int_{0}^{R_{\Omega}}g(t)\mu^{\prime}_{f}(t)\,dt=\int_{0}^{R_{\Omega}}g^{\prime}(t)\mu_{f}(t)\,dt.$

Consequently, (29) becomes

$T_{f,p}(\Omega)\geq\frac{\displaystyle{\left(\int_{0}^{R_{\Omega}}g^{\prime}(t)\mu_{f}(t)\,dt\right)^{\frac{p}{p-1}}}}{\left(\displaystyle{\int_{0}^{R_{\Omega}}g^{\prime
p}(t)P(t)\,dt}\right)^{\frac{1}{p-1}}}.$

We can choose

$g(t)=\int_{0}^{t}\left(\dfrac{\mu_{f}(s)}{P(s)}\right)^{1/(p-1)}ds$

and we observe that $g\in W^{1,p}([0,R_{\Omega}])$, since, using (31), we have

$\displaystyle
g(t)\leq\int_{0}^{R_{\Omega}}(R_{\Omega}-s)^{\frac{1}{p-1}}f(s)^{\frac{1}{p-1}}\,ds\leq{\left\|f\right\|}_{L^{\infty}}^{\frac{1}{p-1}}R_{\Omega}^{\frac{p}{p-1}}\in
L^{p}([0,R_{\Omega}]),$ $\displaystyle
g^{\prime}(t)\leq{\left\|f\right\|}_{L^{\infty}}^{\frac{1}{p-1}}R_{\Omega}^{\frac{1}{p-1}}\in
L^{p}([0,R_{\Omega}]).$

So, we have

$T_{f,p}(\Omega)\geq\int_{0}^{R_{\Omega}}\frac{\mu_{f}^{\frac{p}{p-1}}(t)}{P^{\frac{1}{p-1}}(t)}\,dt=-\frac{p-1}{2p-1}\int_{0}^{R_{\Omega}}\frac{(\mu^{\frac{2p-1}{p-1}}_{f}(t))^{\prime}}{f(t)P^{\frac{p}{p-1}}(t)}\,dt.$
(33)

Let us set $c_{p}=(p-1)/(2p-1)$. Since $f(s)$ is a non-negative and non-
increasing function, integrating by parts in (33), we get

$\displaystyle T_{f,p}(\Omega)$ $\displaystyle\geq-
c_{p}\int_{0}^{R_{\Omega}}\frac{(\mu^{\frac{2p-1}{p-1}}_{f}(t))^{\prime}}{f(t)P^{\frac{p}{p-1}}(t)}\,dt=-c_{p}\frac{\mu^{\frac{2p-1}{p-1}}_{f}(t)}{f(t)P^{\frac{p}{p-1}}(t)}\Bigg{\rvert}_{0}^{R_{\Omega}}+$
(34) $\displaystyle-
c_{p}\int_{0}^{R_{\Omega}}\frac{\mu^{\frac{2p-1}{p-1}}_{f}(t)}{f^{2}(t)P^{\frac{2p}{p-1}}(t)}\left(f^{\prime}(t)P^{\frac{p}{p-1}}(t)+\frac{p}{p-1}f(t)P^{\frac{1}{p-1}}(t)P^{\prime}(t)\right)\,dt$
$\displaystyle\geq
c_{p}\frac{\mu^{\frac{2p-1}{p-1}}_{f}(\Omega)}{f(0)P^{\frac{p}{p-1}}(\Omega)}+\frac{c_{p}}{P^{\frac{p}{p-1}}(\Omega)}\int_{0}^{R_{\Omega}}\frac{\mu^{\frac{2p-1}{p-1}}_{f}(t)}{f^{2}(t)}(-f^{\prime}(t))\,dt,$

where in the last inequality we use (31) and the fact that $P^{\prime}(t)\leq
0$. Now, since $f(s)$ is non-increasing, we obtain the desired estimate

$T_{f,p}(\Omega)\geq
c_{p}\frac{\mu^{\frac{2p-1}{p-1}}_{f}(\Omega)}{f(0)P^{\frac{p}{p-1}}(\Omega)}.$
(35)

∎

#### Step 2: proof of the sharpness of (10)

###### Proof.

We prove that inequality (10) is sharp and that the optimum is asymptotically
achieved by the sequence of thinning cylinders $\Omega_{l}$ with unitary
measure, as defined in (9), that is

$\Omega_{l}=l^{-\frac{1}{n-1}}C\times\left[-\frac{l}{2},\frac{l}{2}\right]$

where $C\subseteq\operatorname{\mathbb{R}}^{n-1}$ is a bounded, open and
convex set with unitary $n-1$-measure. It is easy to verify that, for $n\geq
3$,

$\displaystyle P(\Omega_{l})$
$\displaystyle=2\mathcal{H}^{n-1}(l^{-\frac{1}{n-1}}C)+l\mathcal{H}^{n-2}(\partial(l^{-\frac{1}{n-1}}C))$
(36) $\displaystyle=2l^{-1}+l^{\frac{1}{n-1}}\mathcal{H}^{n-2}(\partial C),$

and we observe that, in the case $n=2$, we have that
$\mathcal{H}^{n-2}(\partial C)=2$.

Let $u$ be the solution to the following $p$-torsion problem

$\begin{cases}-\Delta_{p}u=1&\text{in }\Omega_{l}\\\ u=0&\text{on
}\partial\Omega_{l},\end{cases}$

such that

$\int_{\Omega_{l}}u\;dx=T_{p}(\Omega_{l}),$

and let us consider the following function, depending only on the last
component $x_{n}$ of $x\in\operatorname{\mathbb{R}}^{n}$,

$v(x)=\frac{p-1}{p}\left[\left(\frac{l}{2}\right)^{\frac{p}{p-1}}-{\left|x_{n}\right|}^{\frac{p}{p-1}}\right],$

satisfying

$\begin{cases}-\Delta_{p}v=1&\text{in }\Omega_{l}\\\ v\geq 0&\text{on
}\partial\Omega_{l}.\end{cases}$

The comparison principle, see [14], ensures that $u\leq v$ in $\Omega_{l}$
and, as a consequence,

$\displaystyle T_{p}(\Omega_{l})$
$\displaystyle=\int_{\Omega_{l}}u\,dx\leq\int_{\Omega_{l}}v\,dx=$ (37)
$\displaystyle=\frac{p-1}{p}\int_{l^{-\frac{1}{n-1}}C}\int_{-\frac{l}{2}}^{\frac{l}{2}}\left[\left(\frac{l}{2}\right)^{\frac{p}{p-1}}-{\left|x_{n}\right|}^{\frac{p}{p-1}}\right]\,dx_{n}\,d\mathcal{H}^{n-1}$
$\displaystyle=2\;\frac{p-1}{p}l^{-1}\int_{0}^{\frac{l}{2}}\left[\left(\frac{l}{2}\right)^{\frac{p}{p-1}}-x_{n}^{\frac{p}{p-1}}\right]\,dx_{n}$
$\displaystyle=2\;\frac{p-1}{p}\left[1-\frac{p-1}{2p-1}\right]l^{-1}\left(\frac{l}{2}\right)^{\frac{2p-1}{p-1}}=2c_{p}l^{-1}\left(\frac{l}{2}\right)^{\frac{2p-1}{p-1}}.$

By (37) and (36), we have

$T_{p}(\Omega_{l})P^{\frac{p}{p-1}}(\Omega_{l})\leq
2c_{p}l^{-1}\left(\frac{l}{2}\right)^{\frac{2p-1}{p-1}}\left(2l^{-1}+l^{\frac{1}{n-1}}\mathcal{H}^{n-2}(\partial
C)\right)^{\frac{p}{p-1}}=c_{p}\left(1+\frac{l^{\frac{n}{n-1}}}{2}\mathcal{H}^{n-2}(\partial
C)\right)^{\frac{p}{p-1}}.$

Now, since $f(x)\leq f(0)$, we have that, for every bounded, open and convex
set $\Omega$,

$T_{f,p}(\Omega)\leq f^{\frac{p}{p-1}}(0)T_{p}(\Omega).$ (38)

It follows that

$\displaystyle T_{f,p}(\Omega_{l})P^{\frac{p}{p-1}}(\Omega_{l})$
$\displaystyle\leq
f^{\frac{p}{p-1}}(0)T_{p}(\Omega_{l})P^{\frac{p}{p-1}}(\Omega_{l})$ (39)
$\displaystyle\leq
c_{p}f^{\frac{p}{p-1}}(0)\left(1+\frac{l^{\frac{n}{n-1}}}{2}\mathcal{H}^{n-2}(\partial
C)\right)^{\frac{p}{p-1}}.$

Moreover we observe that, if $f$ never vanishes, we can use its monotonicity
property to bound $\mu_{f}$ from below in the following way:

$\mu_{f}(\Omega)=\int_{\Omega}f(d(x))\,dx\geq f(R_{\Omega})|\Omega|,$

obtaining

$T_{f,p}(\Omega)\geq
c_{p}\frac{f^{\frac{2p-1}{p-1}}(R_{\Omega})|\Omega|^{\frac{2p-1}{p-1}}}{f(0)P^{\frac{p}{p-1}}(\Omega)}.$
(40)

Joining (40) and (39), we obtain

$c_{p}\frac{f^{\frac{2p-1}{p-1}}(R_{\Omega_{l}})}{f(0)}\leq
T_{f,p}(\Omega_{l})P^{\frac{p}{p-1}}(\Omega_{l})\leq
c_{p}f^{\frac{p}{p-1}}(0)\left(1+\frac{l^{\frac{n}{n-1}}}{2}\mathcal{H}^{n-2}(\partial
C)\right)^{\frac{p}{p-1}}.$

Eventually, passing to the limit when $l\to 0$, observing that
$\displaystyle\lim_{l\to 0}R_{\Omega_{l}}=0$ and that $f$ is continuous, we
have

$T_{f,p}(\Omega_{l})P^{\frac{p}{p-1}}(\Omega_{l})\longrightarrow
c_{p}f^{\frac{p}{p-1}}(0).$

∎

######  Remark 3.1.

If we assume that $f:[0,R_{\Omega}]\to[0,+\infty[$ is a function in
$L^{\infty}([0,R_{\Omega}])$, then, using the variational characterization (2)
and the result (6) proved in [9], we have

$T_{f,p}(\Omega)\geq\left(\inf_{t\in[0,R_{\Omega}]}f(t)\right)^{\frac{p}{p-1}}T_{p}(\Omega)\geq\left(\inf_{t\in[0,R_{\Omega}]}f(t)\right)^{\frac{p}{p-1}}c_{p}\dfrac{|\Omega|^{\frac{2p-1}{p-1}}}{P(\Omega)^{\frac{p}{p-1}}}$
(41)

and the sharpness of (41) can be proved in an analogous way as in (10).

##  4 The quantitative results

#### Proof of Theorem 1.2

###### Proof.

If $n=2$ and $f\equiv 1$, (34) becomes

$T_{p}(\Omega)\geq
c_{p}\frac{|\Omega|^{\frac{2p-1}{p-1}}}{P^{\frac{p}{p-1}}(\Omega)}+c_{p}\frac{p}{p-1}\int_{0}^{R_{\Omega}}\left(\frac{\mu(t)}{P(t)}\right)^{\frac{2p-1}{p-1}}(-P^{\prime}(t))\,dt.$
(42)

Joining (17), (18), (26) and (42), we have that

$\displaystyle\frac{T_{p}(\Omega)P^{\frac{p}{p-1}}(\Omega)}{|\Omega|^{\frac{2p-1}{p-1}}}-c_{p}$
$\displaystyle>c_{p}\frac{p}{p-1}\frac{P^{\frac{p}{p-1}}(\Omega)}{|\Omega|^{\frac{2p-1}{p-1}}}\int_{0}^{R_{\Omega}}\left(\frac{\mu(t)}{P(t)}\right)^{\frac{2p-1}{p-1}}(-P^{\prime}(t))\,dt$
$\displaystyle\geq\frac{\pi}{2^{\frac{p}{p-1}}}\frac{p}{2p-1}\frac{P^{\frac{p}{p-1}}(\Omega)}{|\Omega|^{\frac{2p-1}{p-1}}}\int_{0}^{R_{\Omega}}\left(R_{\Omega}-t\right)^{\frac{2p-1}{p-1}}\,dt$
$\displaystyle\geq\frac{\pi}{2^{\frac{p}{p-1}}}\frac{(p-1)p}{(3p-2)(2p-1)}\frac{R_{\Omega}}{P(\Omega)}\left(\frac{R_{\Omega}P(\Omega)}{|\Omega|}\right)^{{\frac{2p-1}{p-1}}}$
$\displaystyle\geq\frac{\pi}{2^{\frac{p}{p-1}}}\frac{(p-1)p}{(3p-2)(2p-1)}\frac{R_{\Omega}}{P(\Omega)}.$

Hence, by applying (21) and (18) we get

$\mathcal{F}_{p}(\Omega)-c_{p}\geq
K(p)\frac{w_{\Omega}}{\operatorname{\,\textup{diam}}(\Omega)},$ (43)

where

$K(p):=\frac{(p-1)p}{2^{\frac{p}{p-1}}3(3p-2)(2p-1)}.$

We now prove that the exponent of this ratio is sharp. In order to do that,
since we have proved (43), we only need to find a sequence
$\\{\Omega_{l}\\}_{l\in\mathbb{N}}$ of convex sets with fixed measure such
that

$M\frac{w_{\Omega_{l}}}{\operatorname{\,\textup{diam}}(\Omega_{l})}\geq\mathcal{F}_{p}(\Omega_{l})-c_{p},$

for some positive constant $M$. Let $0<l<1$, we consider the following
rectangle

$\Omega_{l}=\left(-\frac{1}{2l},\frac{1}{2l}\right)\times\left(-\frac{l}{2},\frac{l}{2}\right)$

and we notice that its inradius and area are $R_{\Omega_{l}}=\frac{l}{2}$ and
${\left|\Omega_{l}\right|}=1$. Let $u$ be the unique solution to

$\begin{cases}-\Delta_{p}u=1\quad\text{in }\Omega_{l}\\\ u=0\qquad\text{on
}\partial\Omega_{l}\end{cases}$

and let us consider the following function

$v(y)=\frac{p-1}{p}\left[\left(\frac{l}{2}\right)^{\frac{p}{p-1}}-{\left|y\right|}^{\frac{p}{p-1}}\right],$

which solves

$\begin{cases}-\Delta v=1&\text{in }\Omega_{l}\\\ v\geq 0&\text{on
}\partial\Omega_{l}.\end{cases}$

The comparison principle gives $u\leq v$ in $\Omega_{l}$ and

$T_{p}(\Omega_{l})=\int_{\Omega_{l}}u_{p}\,dx\leq\int_{\Omega_{l}}v\,dx.$

Arguing as in (37), we have

$\int_{\Omega_{l}}v\,dx=c_{p}\left(\frac{l}{2}\right)^{\frac{p}{p-1}}.$

On the other hand, the perimeter of the rectangle is given by

$P(\Omega_{l})=\frac{2}{l}\left(1+l^{2}\right)$

and its Taylor expansion with respect to $l>0$ is

$P^{{\frac{p}{p-1}}}(\Omega_{l})=\left(\frac{2}{l}\right)^{{\frac{p}{p-1}}}\left(1+l^{2}\right)^{{\frac{p}{p-1}}}=\left(\frac{2}{l}\right)^{{\frac{p}{p-1}}}\left(1+\frac{p}{p-1}l^{2}+o(l^{2})\right).$

Using (18) and (21), we get

$\displaystyle T_{p}(\Omega_{l})P^{\frac{p}{p-1}}(\Omega_{l})-c_{p}$
$\displaystyle\leq
c_{p}\left(\frac{l}{2}\right)^{\frac{p}{p-1}}\left(\frac{2}{l}\right)^{{\frac{p}{p-1}}}\left(1+\frac{p}{p-1}l^{2}+o(l^{2})\right)-c_{p}$
$\displaystyle\leq 2c_{p}\frac{p}{p-1}l^{2}\leq
16c_{p}\frac{p}{p-1}\frac{R_{\Omega_{l}}}{P(\Omega_{l})}$ $\displaystyle\leq
4c_{p}\frac{p}{p-1}\frac{w_{\Omega_{l}}}{\operatorname{\,\textup{diam}}(\Omega_{l})}$

and this concludes the proof. ∎

In the following Remark, we give a generalization of inequality (12) in
dimension $n\geq 2$.

######  Remark 4.1.

It is possible to generalize the main result of Theorem 1.2, obtaining that,
for $\Omega$ bounded, open and convex, we have

$\frac{T_{p}(\Omega)P^{\frac{p}{p-1}}(\Omega)}{|\Omega|^{\frac{2p-1}{p-1}}}-c_{p}\geq
C(n,p)\bigg{(}\frac{w_{\Omega}}{\operatorname{\,\textup{diam}}(\Omega)}\bigg{)}^{n-1}.$
(44)

Indeed, if we choose $f\equiv 1$, (34) becomes

$T_{p}(\Omega)\geq
c_{p}\frac{|\Omega|^{\frac{2p-1}{p-1}}}{P^{\frac{p}{p-1}}(\Omega)}+c_{p}\frac{p}{p-1}\int_{0}^{R_{\Omega}}\left(\frac{\mu(t)}{P(t)}\right)^{\frac{2p-1}{p-1}}(-P^{\prime}(t))\,dt.$
(45)

As a consequence of the Alexandrov-Fenchel inequality and the isoperimetric
inequality for the quermassintegrals (see [20]), we have

$-P^{\prime}(t)\geq
n(n-1)\omega_{n}^{\frac{1}{n-1}}\bigg{(}\frac{P(t)}{n}\bigg{)}^{\frac{n-2}{n-1}}.$
(46)

Hence, combining (46) and (45), we have

$\frac{T_{p}(\Omega)P^{\frac{p}{p-1}}(\Omega)}{|\Omega|^{\frac{2p-1}{p-1}}}-c_{p}\geq
c(n,p)\frac{P^{\frac{p}{p-1}}(\Omega)}{|\Omega|^{\frac{2p-1}{p-1}}}\int_{0}^{R_{\Omega}}\left(\frac{\mu(t)}{P(t)}\right)^{\frac{2p-1}{p-1}}P(t)^{\frac{n-2}{n-1}}\,dt.$
(47)

Moreover, from (17), we obtain that

$P(t)\geq n\omega_{n}(R_{\Omega}-t)^{n-1},$ (48)

and so, using (48) in (47), we get

$\begin{split}\frac{T_{p}(\Omega)P^{\frac{p}{p-1}}(\Omega)}{|\Omega|^{\frac{2p-1}{p-1}}}-c_{p}&\geq
c(n,p)\frac{P^{\frac{p}{p-1}}(\Omega)}{|\Omega|^{\frac{2p-1}{p-1}}}\int_{0}^{R_{\Omega}}\left(R_{\Omega}-t\right)^{\frac{2p-1}{p-1}+n-2}\,dt\\\
&=C(n,p)\bigg{(}\frac{R_{\Omega}P(\Omega)}{|\Omega|}\bigg{)}^{\frac{2p-1}{p-1}}\frac{R_{\Omega}^{n-1}}{P(\Omega)}\end{split}$
(49)

If we combine (49) with (17), with the following estimate (that can be found
in [3]):

$\displaystyle
R_{\Omega}\geq\begin{cases}w_{\Omega}\displaystyle{\frac{\sqrt{n+2}}{2n+2}}&n\,\,\text{even}\\\
\\\ w_{\Omega}\displaystyle{\frac{1}{2\sqrt{n}}}&n\,\,\text{odd},\end{cases}$

and with

$\displaystyle{P(\Omega)\leq
n\omega_{n}\left(\frac{n}{2n+2}\right)^{\frac{n-1}{2}}\operatorname{\,\textup{diam}}(\Omega)^{n-1}},$

we finally get

$\frac{T_{p}(\Omega)P^{\frac{p}{p-1}}(\Omega)}{|\Omega|^{\frac{2p-1}{p-1}}}-c_{p}\geq
C(n,p)\bigg{(}\frac{w_{\Omega}}{\operatorname{\,\textup{diam}}(\Omega)}\bigg{)}^{n-1}.$

As far as the sharpness of (12), we conjecture that the sharp exponent is $1$
as in the planar case. Indeed, the minimizing sequence $\\{\Omega_{l}\\}$
satisfies

$T_{p}(\Omega_{l})P^{\frac{p}{p-1}}(\Omega_{l})-c_{p}\approx
C\frac{w_{\Omega_{l}}}{\operatorname{\,\textup{diam}}(\Omega_{l})}.$

######  Remark 4.2.

As already remarked in the introduction, inequality (43) gives an information
on the set $\Omega$. Indeed, if

$\mathcal{F}_{p}(\Omega)-c_{p}$

is small, then the ratio between $w_{\Omega}$ and
$\operatorname{\,\textup{diam}}(\Omega)$ has to be necessarily small, i.e.
$\Omega$ must be a thin domain. Moreover, inequality (43) tells us also that
the minimum of $\mathcal{F}_{p}(\Omega)$ is not achieved among bounded, open
and convex sets. Assuming by contradiction that there exists a bounded, open
and convex set $\tilde{\Omega}$ such that

$\mathcal{F}_{p}(\tilde{\Omega})=c_{p},$

we have that

$\frac{w_{\tilde{\Omega}}}{\operatorname{\,\textup{diam}}({\tilde{\Omega}})}<\varepsilon\,\,\,\,\,\,\,\,\,\,\,\forall\varepsilon>0,$

which is an absurd.

Theorem 1.2 only tells us that any minimizing sequence of
$\mathcal{F}_{p}(\cdot)$ is a sequence of thinning domains. On the other hand,
Theorem 1.3 gives us more precise information on the geometry of such
minimizing sequence.

#### Proof of Theorem 1.3

###### Proof.

Let $\Omega$ be a bounded, open and convex set with nonempty interior in
$\mathbb{R}^{2}$ and let us consider a rectangle $Q$ of sides $P(\Omega)/2$
and $w_{\Omega}$ containing $\Omega$. Such a rectangle exists, since it is
enough to choose the shorter side of $Q$ parallel to the direction of
$w_{\Omega}$ and to recall the lower bound in (21) (see Figure 4).

$\operatorname{\,\textup{diam}}(\Omega)$$w_{\Omega}$$\Omega$$P(\Omega)/2$

Now, let $\sigma>0$ be such that

$\displaystyle\frac{1}{4^{3}\cdot 6}-\frac{\pi^{2}}{2^{3}\cdot
3^{3}}\frac{\sigma^{2}}{K^{2}(2)}\geq 0;$ (50)
$\displaystyle\frac{1}{3^{3}\cdot
6}-\frac{\pi}{48}\frac{\sigma}{K(2))}-\frac{\pi^{2}}{2^{5}\cdot
3}\frac{\sigma^{2}}{K^{2}(2)}\geq 0;$ (51)
$\displaystyle\frac{\pi}{4}-\frac{\pi}{2\sqrt{3}}\frac{\sigma}{K(2)}\geq\frac{4}{3\sqrt{3}},$
(52)

where $K(2)$ is a constant defined in (13). If

$\frac{T(\Omega)P^{2}(\Omega)}{|\Omega|^{3}}-\frac{1}{3}\geq\sigma,$

then, by (18) and (17), we have

$\frac{{\left|Q\bigtriangleup\Omega\right|}}{{\left|\Omega\right|}}=\left(\frac{P(\Omega)w_{\Omega}}{2{\left|\Omega\right|}}-1\right)\leq\left(\frac{3}{2}\frac{P(\Omega)R_{\Omega}}{{\left|\Omega\right|}}-1\right)\leq
2.$

So, it follows that

$\frac{T(\Omega)P^{2}(\Omega)}{|\Omega|^{3}}-\frac{1}{3}\geq\frac{\sigma}{2^{3}}2^{3}\geq\frac{\sigma}{2^{3}}\left(\frac{{\left|Q\bigtriangleup\Omega\right|}}{|\Omega|}\right)^{3}.$

On the other hand, let us assume that

$\frac{T(\Omega)P^{2}(\Omega)}{|\Omega|^{3}}-\frac{1}{3}<\sigma.$ (53)

By Theorem 1.2, we have that

$\frac{w_{\Omega}}{\operatorname{\,\textup{diam}}(\Omega)}\leq\frac{1}{K(2)}\left[\frac{T(\Omega)P^{2}(\Omega)}{|\Omega|^{3}}-\frac{1}{3}\right]<\frac{\sigma}{K(2)},$
(54)

and we observe that, by the choice of $\sigma$ made in (50)-(52), a ball
cannot satisfy (53).

Now, arguing as in (33) with $f\equiv 1$ and $p=2$, we know that

$T(\Omega)\geq\int_{0}^{R_{\Omega}}\frac{\mu^{2}(t)}{P(t)}\,dt.$ (55)

We set $\displaystyle{\rho=\dfrac{P^{2}(\Omega)}{4\pi}-{\left|\Omega\right|}}$
and $p_{R}=P(\Omega)-2\pi R_{\Omega}$ and we observe that they are both
strictly positive by the isoperimetric inequality and the monotonicity of the
perimeter, respectively. Using inequalities (24) and (25) in (55), we have
that

$\begin{split}T(\Omega)P^{2}(\Omega)&\geq
P^{2}(\Omega)\int_{0}^{R_{\Omega}}\frac{(|\Omega|-P(\Omega)t+\pi
t^{2})^{2}}{P(\Omega)-2\pi t}\,dt\\\
&=P^{2}(\Omega)\int_{0}^{R_{\Omega}}\frac{1}{P(\Omega)-2\pi
t}\left(\frac{\left(P(\Omega)-2\pi
t\right)^{2}}{4\pi}-\left(\frac{P^{2}(\Omega)}{4\pi}-{\left|\Omega\right|}\right)\right)^{2}\,dt\\\
&=P^{2}(\Omega)\int_{0}^{R_{\Omega}}\left(\frac{\left(P(\Omega)-2\pi
t\right)^{3}}{(4\pi)^{2}}-\frac{\rho}{2\pi}\left(P(\Omega)-2\pi
t\right)+\frac{\rho^{2}}{P(\Omega)-2\pi t}\right)\,dt\\\
&=\frac{P^{2}(\Omega)}{2\pi}\left(\frac{P^{4}(\Omega)-p_{R}^{4}}{4(4\pi)^{2}}-\frac{\rho}{4\pi}\left(P^{2}(\Omega)-p_{R}^{2}\right)-\rho^{2}\log\left(1-\frac{2\pi
R_{\Omega}}{P(\Omega)}\right)\right),\end{split}$ (56)

and, using Newton’s formula and the Taylor series for the logarithm, we get

$\displaystyle P^{2}(\Omega)-p^{2}_{R}=4\pi
R_{\Omega}P(\Omega)-4\pi^{2}R^{2}_{\Omega};$ (57) $\displaystyle
P^{4}(\Omega)-p^{4}_{R}=8\pi
R_{\Omega}P^{3}(\Omega)-24\pi^{2}R_{\Omega}^{2}P^{2}(\Omega)+32\pi^{3}R^{3}_{\Omega}P(\Omega)-16\pi^{4}R^{4}_{\Omega};$
$\displaystyle-\log\left(1-\frac{2\pi
R_{\Omega}}{P(\Omega)}\right)=\sum_{i=1}^{\infty}\frac{1}{i}\left(\frac{2\pi
R_{\Omega}}{P(\Omega)}\right)^{i}\geq\frac{2\pi
R_{\Omega}}{P(\Omega)}+\frac{2\pi^{2}R^{2}_{\Omega}}{P^{2}(\Omega)}+\frac{8}{3}\frac{\pi^{3}R^{3}_{\Omega}}{P^{3}(\Omega)}+\frac{4\pi^{4}R^{4}_{\Omega}}{P^{4}(\Omega)}.$

By (57) and (56), dividing by ${\left|\Omega\right|}^{3}$ and subtracting
$1/3$, we have

$\displaystyle\frac{T(\Omega)P^{2}(\Omega)}{{\left|\Omega\right|}^{3}}-\frac{1}{3}\geq$
$\displaystyle\frac{1}{3}\bigg{(}\frac{P(\Omega)R_{\Omega}}{{\left|\Omega\right|}}-1\bigg{)}^{3}+\pi\frac{R_{\Omega}^{2}}{{\left|\Omega\right|}^{2}}\left({\left|\Omega\right|}-\frac{2}{3}P(\Omega)R_{\Omega}\right)$
(58)
$\displaystyle+\frac{4}{3}\pi^{2}\frac{R_{\Omega}^{3}}{P(\Omega){\left|\Omega\right|}^{2}}\left({\left|\Omega\right|}-\frac{3}{4}P(\Omega)R_{\Omega}\right).$

As an intermediate step we want to prove the following inequality:

$\begin{multlined}\frac{1}{3}\bigg{(}\frac{P(\Omega)R_{\Omega}}{{\left|\Omega\right|}}-1\bigg{)}^{3}+\pi\frac{R_{\Omega}^{2}}{{\left|\Omega\right|}^{2}}\left({\left|\Omega\right|}-\frac{2}{3}P(\Omega)R_{\Omega}\right)\\\
+\frac{4}{3}\pi^{2}\frac{R_{\Omega}^{3}}{P(\Omega){\left|\Omega\right|}^{2}}\left({\left|\Omega\right|}-\frac{3}{4}P(\Omega)R_{\Omega}\right)\geq\frac{1}{6}\bigg{(}\frac{P(\Omega)R_{\Omega}}{{\left|\Omega\right|}}-1\bigg{)}^{3},\end{multlined}\frac{1}{3}\bigg{(}\frac{P(\Omega)R_{\Omega}}{{\left|\Omega\right|}}-1\bigg{)}^{3}+\pi\frac{R_{\Omega}^{2}}{{\left|\Omega\right|}^{2}}\left({\left|\Omega\right|}-\frac{2}{3}P(\Omega)R_{\Omega}\right)\\\
+\frac{4}{3}\pi^{2}\frac{R_{\Omega}^{3}}{P(\Omega){\left|\Omega\right|}^{2}}\left({\left|\Omega\right|}-\frac{3}{4}P(\Omega)R_{\Omega}\right)\geq\frac{1}{6}\bigg{(}\frac{P(\Omega)R_{\Omega}}{{\left|\Omega\right|}}-1\bigg{)}^{3},$
(59)

that, combined with (58), implies

$\frac{T(\Omega)P^{2}(\Omega)}{|\Omega|^{3}}-\frac{1}{3}\geq\frac{1}{6}\left(\frac{P(\Omega)R_{\Omega}}{|\Omega|}-1\right)^{3},$
(60)

where we choose the constant $1/6$ as an arbitrary constant less then $1/3$.
In particular, (59) is equivalent to

$\frac{1}{6}\big{(}P(\Omega)R_{\Omega}-|\Omega|\big{)}^{3}+\pi
R_{\Omega}^{2}{\left|\Omega\right|}\left({\left|\Omega\right|}-\frac{2}{3}P(\Omega)R_{\Omega}\right)+\frac{4}{3}\pi^{2}\frac{R_{\Omega}^{3}}{P(\Omega)}{\left|\Omega\right|}\left({\left|\Omega\right|}-\frac{3}{4}P(\Omega)R_{\Omega}\right)\geq
0.$ (61)

In order to prove (61), we distinguish three cases:

  1. 1)

if $\displaystyle{{\left|\Omega\right|}\geq\frac{3}{4}P(\Omega)R_{\Omega}}$,
then (61) is trivial, since the left hand side is the sum of positive
quantities;

  2. 2)

if
$\displaystyle{\frac{2}{3}P(\Omega)R_{\Omega}\leq{\left|\Omega\right|}<\frac{3}{4}P(\Omega)R_{\Omega}}$,
using (18), (21), (50) and (54), we have

$\begin{split}\frac{1}{6}\big{(}P(\Omega)R_{\Omega}-|\Omega|\big{)}^{3}&+\pi
R_{\Omega}^{2}{\left|\Omega\right|}\left({\left|\Omega\right|}-\frac{2}{3}P(\Omega)R_{\Omega}\right)+\frac{4}{3}\pi^{2}\frac{R_{\Omega}^{3}}{P(\Omega)}{\left|\Omega\right|}\left({\left|\Omega\right|}-\frac{3}{4}P(\Omega)R_{\Omega}\right)\\\
&\geq P^{3}(\Omega)R^{3}_{\Omega}\left(\frac{1}{4^{3}\cdot
6}-\frac{2\pi^{2}}{3^{3}}\frac{R^{2}_{\Omega}}{P^{2}(\Omega)}\right)\\\ &\geq
P^{3}(\Omega)R^{3}_{\Omega}\left(\frac{1}{4^{3}\cdot
6}-\frac{\pi^{2}}{2^{3}\cdot
3^{3}}\frac{w^{2}_{\Omega}}{\operatorname{\,\textup{diam}}^{2}(\Omega)}\right)\\\
&\geq P^{3}(\Omega)R^{3}_{\Omega}\left(\frac{1}{4^{3}\cdot
6}-\frac{\pi^{2}}{2^{3}\cdot 3^{3}}\frac{\sigma^{2}}{K^{2}(2)}\right)\geq
0.\end{split}$ (62)

  3. 3)

if
$\displaystyle{\frac{1}{2}P(\Omega)R_{\Omega}\leq{\left|\Omega\right|}<\frac{2}{3}P(\Omega)R_{\Omega}}$,
arguing as before, we have

$\begin{split}\frac{1}{6}\big{(}P(\Omega)R_{\Omega}-|\Omega|\big{)}^{3}&+\pi
R_{\Omega}^{2}{\left|\Omega\right|}\left({\left|\Omega\right|}-\frac{2}{3}P(\Omega)R_{\Omega}\right)+\frac{4}{3}\pi^{2}\frac{R_{\Omega}^{3}}{P(\Omega)}{\left|\Omega\right|}\left({\left|\Omega\right|}-\frac{3}{4}P(\Omega)R_{\Omega}\right)\\\
&\geq P^{3}(\Omega)R^{3}_{\Omega}\left(\frac{1}{3^{3}\cdot
6}-\frac{\pi}{48}\frac{w_{\Omega}}{\operatorname{\,\textup{diam}}(\Omega)}-\frac{\pi^{2}}{2^{5}\cdot
3}\frac{w^{2}_{\Omega}}{\operatorname{\,\textup{diam}}^{2}(\Omega)}\right)\\\
&\geq P^{3}(\Omega)R^{3}_{\Omega}\left(\frac{1}{3^{3}\cdot
6}-\frac{\pi}{48}\frac{\sigma}{K(2)}-\frac{\pi^{2}}{2^{5}\cdot
3}\frac{\sigma^{2}}{K^{2}(2)}\right)\geq 0.\end{split}$ (63)

So, we have proved the intermediate step (60). Now, by combining (60) and
(19), we deduce

$\frac{T(\Omega)P^{2}(\Omega)}{|\Omega|^{3}}-\frac{1}{3}\geq\frac{1}{6}\left[\frac{P(\Omega)R_{\Omega}}{|\Omega|}-1\right]^{3}\geq\frac{1}{6}\left[\frac{P(\Omega)w_{\Omega}}{2|\Omega|}-1-\frac{1}{\sqrt{3}}\frac{w_{\Omega}^{2}}{{\left|\Omega\right|}}\right]^{3}.$
(64)

Using (20), (19), (54) and (52), we have

$\begin{split}\frac{P(\Omega)w_{\Omega}}{2{\left|\Omega\right|}}-1&\geq\frac{P(\Omega)R_{\Omega}}{{\left|\Omega\right|}}-1\geq\pi\frac{R_{\Omega}^{2}}{{\left|\Omega\right|}}\geq\frac{\pi}{{\left|\Omega\right|}}\left(\frac{w_{\Omega}}{2}-\frac{w_{\Omega}^{2}}{\sqrt{3}P(\Omega)}\right)^{2}\\\
&=\frac{w_{\Omega}^{2}}{{\left|\Omega\right|}}\left(\frac{\pi}{4}-\frac{\pi}{\sqrt{3}}\frac{w_{\Omega}}{P(\Omega)}+\frac{\pi}{3}\frac{w_{\Omega}^{2}}{P(\Omega)^{2}}\right)\\\
&\geq\frac{w_{\Omega}^{2}}{{\left|\Omega\right|}}\left(\frac{\pi}{4}-\frac{\pi}{2\sqrt{3}}\frac{w_{\Omega}}{\operatorname{\,\textup{diam}}(\Omega)}\right)\\\
&\geq\frac{w_{\Omega}^{2}}{{\left|\Omega\right|}}\left(\frac{\pi}{4}-\frac{\pi}{2\sqrt{3}}\frac{\sigma}{K(2)}\right)\\\
&\geq\frac{4}{3\sqrt{3}}\frac{w_{\Omega}^{2}}{{\left|\Omega\right|}}\end{split}$
(65)

Finally, by combining (64) and (65), we get the conclusion

$\frac{T(\Omega)P^{2}(\Omega)}{|\Omega|^{3}}-\frac{1}{3}\geq\frac{1}{6}\left[\frac{P(\Omega)R_{\Omega}}{{\left|\Omega\right|}}-1\right]^{3}\geq\tilde{K}\left[\frac{{\left|Q\bigtriangleup\Omega\right|}}{{\left|\Omega\right|}}\right]^{3}.$
(66)

∎

The next remark shows that a sequence of thinning triangles is not sharp for
(12) and this is the reason for which we need Theorem 1.3 to obtain more
precise information.

######  Remark 4.3.

Let us consider a sequence of isosceles triangles $\mathcal{T}_{l}$ of base
$L$ and height $l$ such that ${\left|\mathcal{T}_{l}\right|}=1$.

$t$$l$$L$$a$$\frac{\theta}{2}$$\frac{\pi}{2}-\theta$$\mathcal{T}_{l}$

If we compute (60) on the sequence $\mathcal{T}_{l}$ and we use (17), we get,
for every $l$,

$\displaystyle\dfrac{T(\mathcal{T}_{l})P^{2}(\mathcal{T}_{l})}{|\mathcal{T}_{l}|^{3}}-\frac{1}{3}\geq\frac{1}{6}\left(\frac{P(\mathcal{T}_{l})R_{\mathcal{T}_{l}}}{{\left|\mathcal{T}_{l}\right|}}-1\right)^{3}=\frac{1}{6}$
(67)

and, so, the quantity on the left-hand side of (67) is bounded away from zero.

######  Remark 4.4.

We point out that

$\frac{P(\Omega)R_{\Omega}}{{\left|\Omega\right|}}-1\geq
K\frac{{\left|Q\bigtriangleup\Omega\right|}}{{\left|\Omega\right|}},$

in (66) is a quantitative version of the inequality in the right hand side of
(17).

######  Open problem 4.5.

We conclude by listing the following open problems:

  * •

Inequality (43) leads us to think that the exponent 3 in (14) is not sharp: we
expect it to be 1.

  * •

We are able to generalize the estimate in Theorem 1.2 in any dimension (see
Remark 4.1 ). We conjecture that the sharp exponent in (44) is $1$.

  * •

The results contained in Theorem 1.3 could be studied in higher dimension and
extended to the $f,p$-torsional rigidity. Our proof cannot be adapted to
higher dimension because in dimension $n>2$ we do not have any more Steiner
formulas for inner parallel sets (24) and (25).

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Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli
studi di Napoli Federico II, Via Cintia, Complesso Universitario Monte S.
Angelo, 80126 Napoli, Italy.

E-mail address, V. Amato<EMAIL_ADDRESS>
E-mail address, A.L. Masiello<EMAIL_ADDRESS>
E-mail address, G. Paoli (Corresponding author<EMAIL_ADDRESS>
E-mail address, R. Sannipoli<EMAIL_ADDRESS>