# On $\pi$-compatible topologies and their special cases

Vitalij A. Chatyrko

###### Abstract

Topologies $\tau,\sigma$ on a set $X$ are called $\pi$-compatible if $\tau$ is
a $\pi$-network for $\sigma$, and vice versa. If topologies $\tau,\sigma$ on a
set $X$ are $\pi$-compatible then the families of nowhere dense sets (resp.
meager sets or sets possessing the Baire property) of the spaces $(X,\tau)$
and $(X,\sigma)$ coincide. A topology $\sigma$ on a set $X$ is called an
admissible extension of a topology $\tau$ on $X$ if $\tau\subseteq\sigma$ and
$\tau$ is a $\pi$-network for $\sigma$. It turns out that examples of
admissible extensions were occurred in literature several times. In the paper
we provide some new facts about the $\pi$-compatibility and the admissible
extension as well as about their particular cases.

Keywords and Phrases: $\pi$-compatible topologies, admissible extension, local
function, Hattori spaces, almost topological groups

2000 AMS (MOS) Subj. Class.: Primary 54A10, 54D10, 54D15

##  1 Introduction

Let $X$ be a non-empty set, $\mathcal{P}(X)$ the family of all subsets of $X$,
$\tau$ a topology on $X$ and $\mathcal{I}_{m}(X,\tau)$ the family of all
meager sets of the topological space $(X,\tau)$.

An interesting collection of subsets of $X$ extending $\tau$ as well as
$\mathcal{I}_{m}(X,\tau)$, is the family $\mathcal{B}_{p}(X,\tau)$ of all
subsets of $X$ possessing the Baire property in $(X,\tau)$. Recall that a
subset $A$ of $X$ has the Baire property in the space $(X,\tau)$ if
$A=(O\setminus M)\cup N$, where $O\in\tau$ and
$M,N\in\mathcal{I}_{m}(X,\tau)$. It is well known (cf. [Ku]) that the family
$\mathcal{B}_{p}(X,\tau)$ is a $\sigma$-algebra of sets which is invariant
under automorphisms of the space $(X,\tau)$.

Let us note that $\mathcal{B}_{p}(X,\tau_{\mbox{triv}})=\\{\emptyset,X\\}$,
$\mathcal{B}_{p}(X,\tau_{\mbox{dis}})=\mathcal{P}(X)$, where
$\tau_{\mbox{triv}}$ (resp. $\tau_{\mbox{dis}}$) is the trivial (resp.
discrete) topology on the set $X$. It is easy to see that for any set $X$ with
$|X|>1$ there is no topology $\tau$ distinct from $\tau_{\mbox{triv}}$ such
that $\mathcal{B}_{p}(X,\tau)=\\{\emptyset,X\\}$. But there are simple
examples of a set $X$ and a topology $\tau$ on $X$ distinct from
$\tau_{\mbox{dis}}$ such that $\mathcal{B}_{p}(X,\tau)=\mathcal{P}(X)$ (see
Corollary 2.1), and a set $Y$ and a topology $\sigma$ on $Y$ distinct from
$\tau_{\mbox{triv}}$ for which $\mathcal{B}_{p}(Y,\sigma)\neq\mathcal{P}(Y)$
(see Example 2.1 and Example 2.2).

Let us recall (cf. [Ku]) that for the real numbers $\mathbb{R}$ with the
Euclidean topology $\tau_{E}$ we have
$\mathcal{B}_{p}(\mathbb{R},\tau_{E})\neq\mathcal{P}(\mathbb{R})$, and there
is a lot of information about elements of the family
$\mathcal{P}(\mathbb{R})\setminus\mathcal{B}_{p}(\mathbb{R},\tau_{E})$ (see
for example [Kh]). It would be interesting to know for what topologies $\tau$
on $\mathbb{R}$ the equality
$\mathcal{B}_{p}(\mathbb{R},\tau)=\mathcal{B}_{p}(\mathbb{R},\tau_{E})$ is
valid.

One can pose more general questions.

###### Question 1.1

Let $X$ be a set and $\tau$ be a topology on $X$.

  * (a)

Describe elements of the family
$\mathcal{P}(X)\setminus\mathcal{B}_{p}(X,\tau)$.

  * (b)

For what topologies $\sigma$ on the set $X$ does the equality
$\mathcal{B}_{p}(X,\tau)=\mathcal{B}_{p}(X,\sigma)$ hold?

The $\pi$-compatibility of topologies $\sigma$ and $\tau$ on a set $X$ (see
Definition 3.1) introduced in [CN1] implies
$\mathcal{I}_{m}(X,\tau)=\mathcal{I}_{m}(X,\sigma)$ (see Lemma 3.1) and
$\mathcal{B}_{p}(X,\tau)=\mathcal{B}_{p}(X,\sigma)$ (see Corollary 3.1) but it
is not equivalent to the equalities (see Example 3.1). It turns out that a
stronger version of $\pi$-compatibility between two topologies, the notion of
the admissible extension (see Definition 4.1), was occurred in literature
several times ( see [H], [F], [CN2] and others). In the paper we provide some
new facts about these relations between topologies which will be valid for the
mentioned cases.

For standard notions we refer to [Ku] and [E].

##  2 Some simple answers to Question 1.1.(a)

###### Proposition 2.1

Let $X$ be a countable set, $Y\subseteq X$ and $\tau$ a topology on $X$. Let
also for each point $x\in Y$ the set $\\{x\\}$ is a nowhere dense set in the
space $(X,\tau)$, and for each point $x\in X\setminus Y$ the set $\\{x\\}$ is
an open set of the space $(X,\tau)$. Then
$\mathcal{I}_{m}(X,\tau)=\mathcal{P}(Y)$,
$\mathcal{B}_{p}(X,\tau)=\mathcal{P}(X)$ and so
$\mathcal{P}(X)\setminus\mathcal{B}_{p}(X,\tau)=\emptyset.$ $\Box$

###### Corollary 2.1

  * (a)

Let $\mathbb{Q}$ be the set of rational numbers with the Euclidean topology
$\tau_{E}$. Then we have
$\mathcal{I}_{m}(\mathbb{Q},\tau_{E})=\mathcal{B}_{p}(\mathbb{Q},\tau_{E})=\mathcal{P}(\mathbb{Q})$.

  * (b)

Let $\mathbb{Z}$ be the set of integer numbers, $\mathbb{Z}_{e}$ be the set of
even integer numbers and $\tau_{K}$ be a topology generated by the family
$\\{\\{n-1,n,n+1\\}:n\in\mathbb{Z}_{e}\\}$ (Khalimsky topology). Then we have
$\mathcal{I}_{m}(\mathbb{Z},\tau_{K})=\mathcal{P}(\mathbb{Z}_{e})\neq\mathcal{P}(\mathbb{Z})=\mathcal{B}_{p}(\mathbb{Z},\tau_{K})$.
$\Box$

The condition ”each one-point set is either nowhere dense or open” in
Proposition 2.1 is essential. In fact,

###### Example 2.1

Let $\mathbb{N}$ be the set of positive integers and $\tau_{p}$ be the odd-
even topology generated by the partition $\mathcal{P}=\\{\\{2k-1,2k\\}\\}$ of
$\mathbb{N}$ ([SS, Example 6]). Note that the family of nowhere dense sets
(resp. meager sets) for the space $(\mathbb{N},\tau_{p})$ is equal to
$\\{\emptyset\\}$. Moreover,
$\mathcal{B}_{p}(\mathbb{N},\tau_{p})=\tau_{p}\neq\mathcal{P}(\mathbb{N})$,
and the family
$\mathcal{P}(\mathbb{N})\setminus\mathcal{B}_{p}(\mathbb{N},\tau_{p})$
consists of sets which are not open.

Let us also note that the condition ”to be countable” on the set $X$ in
Proposition 2.1 is essential.

###### Example 2.2

Let $X$ be any uncountable set and $\tau_{\mbox{fc}}$ be the finite complement
topology on $X$, i.e. $V\in\tau_{\mbox{fc}}$ iff $X\setminus V$ is finite.
Since each infinite subset $Y$ of $X$ is dense in the space
$(X,\tau_{\mbox{fc}})$, $\mathcal{I}_{m}(X,\tau_{\mbox{fc}})=\\{A\subset
X:|A|\leq\aleph_{0}\\}$ and $\mathcal{B}_{p}(X,\tau_{\mbox{fc}})=\\{A\subset
X:|A|\leq\aleph_{0}\mbox{ or }|X\setminus A|\leq\aleph_{0}\\}.$ So any
uncountable subset $Y$ of $X$ such that $X\setminus Y$ is also uncountable,
does not belong to the family $\mathcal{B}_{p}(X,\tau_{\mbox{fc}})$. In
particular, $\mathcal{B}_{p}(X,\tau_{\mbox{fc}})\neq\mathcal{P}(X)$.

##  3 $\pi$-compatible topologies as an answer to Question 1.1.(b).

###### Definition 3.1

([CN1, Definition 3.1]) Let $\tau,\sigma$ be topologies on a set $X$. We call
the topologies $\pi$-compatible if $\tau$ is a $\pi$-network for $\sigma$
(i.e. for each non-empty element $O$ of $\sigma$ there is a non-empty element
$V$ of $\tau$ which is a subset of $O$) and vice versa.

It is evident that the $\pi$-compatibility is an equivalence relation on the
family of all topologies on the set $X$.

###### Lemma 3.1

([CN1, Lemma 3.3]) Let $\tau$ and $\sigma$ be $\pi$-compatible topologies on a
set $X$. Then the spaces $(X,\tau)$ and $(X,\sigma)$ have the same families of
nowhere dense sets (respectively, meager sets). $\Box$

###### Theorem 3.1

Let $\tau$ and $\sigma$ be $\pi$-compatible topologies on a set $X$. Then each
non-empty element of $\tau$ is the union of a non-empty element of $\sigma$
and a nowhere dense set, and vice versa.

Proof. In fact, let $O_{\tau}$ be a non-empty element of $\tau$. Put
$O_{\sigma}=\mbox{{\rm Int}}\,_{\sigma}O_{\tau}$, where $\mbox{{\rm
Int}}\,_{\sigma}A$ (respectively, $\mbox{{\rm Cl}}_{\sigma}A$) is the interior
(respectively, closure) of $A$ in the space $(X,\sigma)$ for any $A\subseteq
X$. Note that $O_{\sigma}\neq\emptyset$ and
$O_{\tau}=O_{\sigma}\cup((\mbox{{\rm Cl}}_{\sigma}O_{\sigma}\setminus
O_{\sigma})\cap O_{\tau})\cup(O_{\tau}\setminus\mbox{{\rm
Cl}}_{\sigma}O_{\sigma})$. Note also that the set $(\mbox{{\rm
Cl}}_{\sigma}O_{\sigma}\setminus O_{\sigma})\cap O_{\tau}$ is nowhere dense.

###### Claim 3.1

The set $A=O_{\tau}\setminus\mbox{{\rm Cl}}_{\sigma}O_{\sigma}$ is nowhere
dense.

Proof. Assume that the set $A$ is not nowhere dense. Put
$W_{\sigma}=\mbox{{\rm Int}}\,_{\sigma}\mbox{{\rm Cl}}_{\sigma}A$. Note that
$W_{\sigma}\neq\emptyset$, $W_{\sigma}\subseteq\mbox{{\rm Cl}}_{\sigma}A$ and
$W_{\sigma}\cap O_{\sigma}=\emptyset$. Since $W_{\sigma}\in\sigma$ and $\tau$
is a $\pi$-network for $\sigma$, there exists an non-empty element $T_{\tau}$
of $\tau$ such that $T_{\tau}\subseteq W_{\sigma}$. There are two
possibilities.

Case 1. $T_{\tau}\cap O_{\tau}\neq\emptyset$. Since $T_{\tau}\cap
O_{\tau}\in\tau$ and $\sigma$ is a $\pi$-network for $\tau$, there is a non-
empty element $P_{\sigma}$ of $\sigma$ such that $P_{\sigma}\subseteq
T_{\tau}\cap O_{\tau}$. Note that $P_{\sigma}\subseteq O_{\tau}\setminus
O_{\sigma}$. We have a contradiction with the definition of $O_{\sigma}$.

Case 2. $T_{\tau}\cap O_{\tau}=\emptyset$. Since $\sigma$ is a $\pi$-network
of $\tau$, there exists a non-empty element $Q_{\sigma}$ of $\sigma$ such that
$Q_{\sigma}\subseteq T_{\tau}$. So $Q_{\sigma}\cap O_{\tau}=\emptyset$ and
hence $Q_{\sigma}\cap\mbox{{\rm Cl}}_{\sigma}O_{\tau}=\emptyset$. But
$Q_{\sigma}\subseteq W_{\sigma}\subseteq\mbox{{\rm
Cl}}_{\sigma}A\subseteq\mbox{{\rm Cl}}_{\sigma}O_{\tau}$. We have a
contradiction.

So $A$ is nowhere dense. $\Box$

By the use of the claim we get that $O_{\tau}$ is the union of $O_{\sigma}$
and the nowhere dense set $((\mbox{{\rm Cl}}_{\sigma}O_{\sigma}\setminus
O_{\sigma})\cap O_{\tau})\cup(O_{\tau}\setminus\mbox{{\rm
Cl}}_{\sigma}O_{\sigma})$. We have done. $\Box$

###### Corollary 3.1

([CN1, Theorem 3.4]) Let $\tau$ and $\sigma$ be $\pi$-compatible topologies on
a set $X$. Then the spaces $(X,\tau)$ and $(X,\sigma)$ have the same families
of Baire sets.

Proof. Let us consider $B\in\mathcal{B}_{p}(X,\tau)$. So $B=(O\setminus M)\cup
N$, where $O$ is an open set in $(X,\tau)$ and $M,N$ are meager sets. We can
assume that $O\neq\emptyset.$ By Theorem 3.1 $O=V\cup A$, where $V$ is a non-
empty open set in $(X,\sigma)$ and $A$ is a nowhere dense set. Note that
$B=((V\cup A)\setminus M)\cup N=(V\setminus M)\cup((A\setminus M)\cup N)$ and
$(A\setminus M)\cup N$ is a meager set. Hence,
$B\in\mathcal{B}_{p}(X,\sigma)$, i. e.
$\mathcal{B}_{p}(X,\tau)\subseteq\mathcal{B}_{p}(X,\sigma)$. The opposite
inclusion can be proved similarly. $\Box$

Besides the $\pi$\- compatible topologies there are other answers to Question
1.1.(b). In fact,

###### Example 3.1

Let $\mathbb{Q}$ be the set of rational numbers, $\tau_{E}$ the Euclidean
topology on $\mathbb{Q}$ and $\tau_{\mbox{fc}}$ be the finite complement
topology on $\mathbb{Q}$. Since each one-point set of spaces
$(\mathbb{Q},\tau_{E})$ and $(\mathbb{Q},\tau_{\mbox{fc}})$ is nowhere dense,
by Proposition 2.1 the families of their meager sets (resp. sets with the
Baire property) are equal to $\mathcal{P}(\mathbb{Q})$. Note that
$\tau_{\mbox{fc}}$ and $\tau_{E}$ are not $\pi$-compatible.

###### Question 3.1

Let $X$ be a set and $\tau,\sigma$ topologies on $X$. What property can we add
to the equality $\mathcal{B}_{p}(X,\tau)=\mathcal{B}_{p}(X,\sigma)$ in order
to get the $\pi$-compatibility of $\tau$ and $\sigma$?

Some weaker property than $\pi$-compatibility, the coincidence of the families
of nowhere dense sets, does not garantee the equality between the families of
the sets with the Baire property. In fact,

###### Example 3.2

Let $\mathbb{N}$ be the set of positive integers, $\tau_{\mbox{dis}}$ be the
discrete topology and $\tau_{p}$ be the odd-even topology (see Example 2.1).
Note that the families of nowhere dense sets (resp. meager sets) for the
spaces $(\mathbb{N},\tau_{\mbox{dis}})$ and $(\mathbb{N},\tau_{p})$ are equal
to $\\{\emptyset\\}$. But
$\mathcal{B}_{p}(\mathbb{N},\tau_{p})=\tau_{p}\neq\mathcal{B}_{p}(\mathbb{N},\tau_{\mbox{dis}})=\mathcal{P}(\mathbb{N})$.

Recall that a topological space $X$ is called a Baire space if for every
countable family $\\{G_{i}\\}_{i=1}^{\infty}$ of open dense subsets of $X$ the
intersection $\cap_{i=1}^{\infty}G_{i}$ is dense in $X$.

It is evident that any open non-empty subset of a Baire space is also Baire.

###### Theorem 3.2

Let $X$ be a set and $\tau,\sigma$ be $\pi$-compatible topologies on $X$. Then
the following is valid.

  * (a)

$(X,\tau)$ is a Baire space iff $(X,\sigma)$ is a Baire space.

  * (b)

Let $\mathcal{G}_{\delta}(X,\tau)$ (resp. $\mathcal{G}_{\delta}(X,\sigma)$ )
be the family of all $G_{\delta}$-sets of the space $(X,\tau)$ (resp.
$(X,\sigma)$).

If $(X,\tau)$ is a Baire space (or $(X,\sigma)$ is a Baire space) then the
family

$\mathcal{B}=\\{\emptyset\neq A\subseteq
X:A\in\mathcal{G}_{\delta}(X,\tau)\cap\mathcal{G}_{\delta}(X,\sigma)\\}$

is a $\pi$-network for the space $(X,\tau)$ and for the space $(X,\sigma)$.

Proof. (a) We show only the necessity.

Let $Y=\cup_{i=1}^{\infty}Y_{i}$, each $Y_{i}$ be closed in the space
$(X,\sigma)$ and $\mbox{{\rm Int}}\,_{\sigma}Y_{i}=\emptyset$. By Lemma 3.1 we
have that $\mbox{{\rm Int}}\,_{\tau}\mbox{{\rm Cl}}_{\tau}Y_{i}=\emptyset$ for
each $i$. Since the space $(X,\tau)$ is Baire, we have $\mbox{{\rm
Int}}\,_{\tau}(\cup_{i=1}^{\infty}\mbox{{\rm Cl}}_{\tau}Y_{i})=\emptyset$.
Assume that $O_{\sigma}=\mbox{{\rm
Int}}\,_{\sigma}(\cup_{i=1}^{\infty}Y_{i})\neq\emptyset$. Note that there is
$\emptyset\neq O_{\tau}\in\tau$ such that $O_{\tau}\subseteq
O_{\sigma}\subseteq\cup_{i=1}^{\infty}Y_{i}.$ Hence,
$O_{\tau}\subseteq\mbox{{\rm Int}}\,_{\tau}(\cup_{i=1}^{\infty}Y_{i})$. Since
$\emptyset\neq\mbox{{\rm
Int}}\,_{\tau}(\cup_{i=1}^{\infty}Y_{i})\subseteq\mbox{{\rm
Int}}\,_{\tau}(\cup_{i=1}^{\infty}\mbox{{\rm Cl}}_{\tau}Y_{i})=\emptyset$, we
have a contradiction.

(b) Let us show that $\mathcal{B}$ is a $\pi$-network for the space
$(X,\tau)$.

Consider a non-empty open set $O_{1}$ in $(X,\tau)$. Apply Theorem 3.1. Choose
a non-empty open set $V_{1}$ in $(X,\sigma)$ such that $V_{1}\subseteq O_{1}$
and $O_{1}\setminus V_{1}$ is a nowhere dense set. Then choose a non-empty
open $O_{2}$ in $(X,\tau)$ such that $O_{2}\subseteq V_{1}$ and
$V_{1}\setminus O_{2}$ is a nowhere dense set, and so on. Since the space
$(X,\tau)$ is a Baire space we have $\cap_{i=1}^{\infty}O_{i}\neq\emptyset$.
Furthermore, by the construction it follows that
$\cap_{i=1}^{\infty}O_{i}=\cap_{i=1}^{\infty}V_{i}\in\mathcal{B}$ and
$\cap_{i=1}^{\infty}O_{i}$ is a dense subset of $O_{1}$. We have done. $\Box$

###### Proposition 3.1

Let $X$ be a set and $\tau,\sigma$ be $\pi$-compatible topologies on $X$. If a
set $A\subset X$ is dense in $(X,\tau)$ then $A$ is dense in $(X,\sigma)$. In
particular, $d(X,\tau)=d(X,\sigma).$ $\Box$

Different examples of $\pi$-compatible topologies one can get combining known
example and the following simple statement.

###### Proposition 3.2

([CN1, Proposition 3.2]) Let $X_{\alpha},\alpha\in\mathcal{A},$ be sets and
for each $\alpha\in\mathcal{A}$ topologies $\tau_{\alpha},\sigma_{\alpha}$ be
$\pi$-compatible on $X_{\alpha}$. Then the topological products of topologies
$\prod\\{\tau_{\alpha},\alpha\in\mathcal{A}\\}$ and
$\prod\\{\sigma_{\alpha},\alpha\in\mathcal{A}\\}$ on the set
$\prod\\{X_{\alpha},\alpha\in\mathcal{A}\\}$ are $\pi$-compatible. $\Box$

Theorem 3.2 and Propositions 3.1, 3.2 easily imply the following statement.

###### Corollary 3.2

Let $\tau_{i}$ and $\sigma_{i}$ be $\pi$-compatible topologies on a set
$X_{i}$ for each $i\leq n$. Then

  * (a)

$d(\prod_{i=1}^{n}(X_{i},\tau_{i}))=d(\prod_{i=1}^{n}(X_{i},\sigma_{i})),$

  * (b)

$\prod_{i=1}^{n}(X_{i},\tau_{i})$ is a Baire space iff
$\prod_{i=1}^{n}(X_{i},\sigma_{i})$ is a Baire space. $\Box$

##  4 Admissible extensions of topologies

###### Definition 4.1

([CN2, Definition 3.1]) A topology $\sigma$ on a set $X$ is said to be an
admissible extension of a topology $\tau$ on the same set $X$ if
$\tau\subseteq\sigma$; and $\tau$ is a $\pi$-base for $\sigma$, i.e. for each
non-empty element $O$ of $\sigma$ there is a non-empty element $V$ of $\tau$
which is a subset of $O$.

Note that if $\tau$ and $\sigma$ are topologies on a set $X$ such that
$\sigma$ is an admissible extension of $\tau$ then $\tau$ and $\sigma$ are
$\pi$-compatible.

###### Theorem 4.1

Let $X$ be a set and $\tau,\sigma$ topologies on $X$ such that $\sigma$ is an
admissible extension of $\tau$. If $O$ is a non-empty element of $\sigma$ then
$O$ is a semi-open set of $(X,\tau)$, i. e. there is an element $V$ of $\tau$
such that $V\subseteq O\subseteq\mbox{{\rm Cl}}_{\tau}V$.

Proof. Put $V=\mbox{{\rm Int}}\,_{\tau}O$ and note that $V\neq\emptyset$. We
will show that $\mbox{{\rm Cl}}_{\tau}V\supseteq O$. In fact, assume that
$W=O\setminus\mbox{{\rm Cl}}_{\tau}V\neq\emptyset.$ Since $\sigma$ is an
admissible extension of $\tau$ then $W\in\sigma$ and there exists
$\emptyset\neq U\in\tau$ such that $U\subseteq W\subseteq O$. It is easy to
see that $U$ must be a subset of $V$. We have a contradiction which proves the
statement. $\Box$

###### Remark 4.1

Theorem 4.1 does not hold for $\pi$-compatible topologies in general. Indeed,
let $\tau_{S}$ be the Sorgenfrey topology on the reals $\mathbb{R}$ generated
by the family $\\{[a,b):a,b\in\mathbb{R}\\}$, $\tau_{-S}$ be a topology on the
reals $\mathbb{R}$ generated by the family $\\{(a,b]:a,b\in\mathbb{R}\\}$ and
$\tau_{E}$ is the natural topology on $\mathbb{R}$. Note that both topologies
$\tau_{S}$ and $\tau_{-S}$ are admissible extensions of $\tau_{E}$. Hence, the
topologies $\tau_{S}$ and $\tau_{-S}$ are $\pi$-compatible. But for a non-
empty element $(a,b]\in\tau_{-S}$ we have the following: $\mbox{{\rm
Int}}\,_{\tau_{S}}(a,b]=(a,b)$ and $\mbox{{\rm
Cl}}_{\tau_{S}}(a,b)=[a,b)\nsupseteq(a,b].$ $\Box$

It is easy to see that the following proposition holds.

###### Proposition 4.1

Let $X$ be a set and $\tau,\sigma$ topologies on $X$ such that $\sigma$ is an
admissible extension of $\tau$. If $(X,\tau)$ is a $T_{0}$-space (resp. a
$T_{1}$-space, a $T_{2}$-space or a functionally Hausdorff space) then
$(X,\sigma)$ is similar. $\Box$

The inverse statement does not hold.

###### Example 4.1

Let $X=X_{0}\cup X_{1}\subset\mathbb{R}^{2}$, where
$X_{i}=\\{\\{x\\}\times\\{i\\}:x\in\mathbb{R}\\}$.

The topology $\sigma$ on $X$ is defined by bases $\mathcal{B}_{\sigma}(p)$ at
each point $p$ of $X$ as follows.

If $p=\\{x\\}\times\\{1\\}$ then
$\mathcal{B}_{\sigma}(p)=\\{[x,y)\times\\{1\\}\cup(x,y)\times\\{0\\}:y>x\\}$.

If $p=\\{x\\}\times\\{0\\}$ then
$\mathcal{B}_{\sigma}(p)=\\{(y,x)\times\\{1\\}\cup(y,x]\times\\{0\\}:y<x\\}$.

The topology $\tau$ on $X$ is also defined by bases $\mathcal{B}_{\tau}(p)$ at
each point $p$ of $X$. Namely,

If $p=\\{x\\}\times\\{1\\}$ or $p=\\{x\\}\times\\{0\\}$ then
$\mathcal{B}_{\tau}(p)=\\{(y,z)\times\\{1\\}\cup(y,z)\times\\{0\\}:y<x<z\\}$.

Let us note that the topology $\sigma$ is an admissible extension of the
topology $\tau$ on the set $X$. Moreover, the space $(X,\sigma)$ is
homeomorphic to a subspace of the Alexandroff double arrow space, and hence
$(X,\sigma)$ is regular $T_{1}$, hereditarily Lindelöf and hereditarily
separable. However, $(X,\tau)$ is not $T_{0}$.

Proposition 4.1 cannot be extended to higher axioms.

###### Example 4.2

([SS, Example 64]) Let us note that there exists an admissible extension
$\sigma$ of the topology $\tau_{E}$ on the reals $\mathbb{R}$ such that
$(\mathbb{R},\sigma)$ is not regular. As $\sigma$ we consider the Smirnov
topology on $\mathbb{R}$: $O\in\sigma$ if $O=U\setminus B$, where
$U\in\tau_{E}$, $B\subseteq A$ and $A=\\{\frac{1}{n}:n=1,2,\dots\\}$.

It is easy to see that if for some topologies $\sigma$ and $\nu$ on a set $X$
there exists a topology $\tau$ on $X$ such that $\sigma$ and $\nu$ are
admissible extensions of $\tau$ then $\nu$ and $\sigma$ are $\pi$-compatible,
and $\tau\subseteq\sigma\cap\nu$.

###### Remark 4.2

We will use the notations from Example 4.1.

Let us consider the topology $\nu$ on $X$ defined by bases
$\mathcal{B}_{\sigma}(p)$ at each point $p$ of $X$ as follows.

If $p=\\{x\\}\times\\{1\\}$ then
$\mathcal{B}_{\sigma}(p)=\\{(y,x)\times\\{0\\}\cup(y,x]\times\\{1\\}:y<x\\}$.

If $p=\\{x\\}\times\\{0\\}$ then
$\mathcal{B}_{\sigma}(p)=\\{[x,y)\times\\{0\\}\cup(x,y)\times\\{1\\}:y>x\\}$.

Note that the topology $\nu$ is an admissible extension of the topology
$\tau$, $\sigma\cap\nu=\tau$ and the spaces $(X,\sigma)$ and $(X,\nu)$ are
homeomorphic.

###### Proposition 4.2

Let $X$ be a set and $\tau,\sigma,\nu$ be topologies on $X$.

  * (a)

If $\sigma\subseteq\nu$ and $\sigma,\nu$ are admissible extensions of $\tau$
then $\nu$ is an admissble extension of $\sigma$.

  * (b)

If $\tau\subseteq\sigma\subseteq\nu$ and $\nu$ ia an admissble extension of
$\tau$ then $\sigma$ is an admissible extension of $\tau$. $\Box$

###### Question 4.1

Let $X$ be a set, $\tau$ and $\sigma$ be $\pi$-compatible topologies on $X$.
Are $\tau$ and $\sigma$ always admissible extensions of $\tau\cap\sigma$?

##  5 Some examples of admissible extensions

###  5.1 Topologies formed from a given topology and ideals of sets

A way to obtain finer topologies (but often non-regular) than a given topology
$\tau$ on a set $X$ is to use ideals of subsets of $X$ (cf. [JH]).

Let $\mathcal{I}$ be an ideal of subsets of $X$ and let $A$ be a subset of
$X$. For a point $x\in X$, let $\mathcal{N}(x)=\\{U\in\tau:x\in U\\}$.

A local function of the set $A$ with the respect to the ideal $\mathcal{I}$
and the topology $\tau$, is the set $A^{*}(\mathcal{I},\tau)=\\{x\in X:A\cap
U\notin\mathcal{I}\text{ for every }U\in\mathcal{N}(x)\\}$.

For every $A\subseteq X$ put $\mbox{{\rm Cl}}^{*}(A)=A\cup
A^{*}(\mathcal{I},\tau)$.

It is easy to see that $\mbox{{\rm Cl}}^{*}(\cdot)$ is a Kuratowski closure
operator. So the family $\tau^{*}(\mathcal{I})=\\{U\subseteq X:\mbox{{\rm
Cl}}^{*}(X\setminus U)=X\setminus U\\}$ is a topology on the set $X$.

Moreover, $\tau\subseteq\tau^{*}(\mathcal{I})$, every element of the ideal
$\mathcal{I}$ is closed in the space $(X,\tau^{*}(\mathcal{I}))$, and
$(\tau^{*}(\mathcal{I}))^{*}(\mathcal{I})=\tau^{*}(\mathcal{I})$.

It is easy to see that if $\mathcal{I}\subseteq\mathcal{J}$ then
$\tau^{*}(\mathcal{I})\subseteq\tau^{*}(\mathcal{J})$.

###### Theorem 5.1

([CN1, Theorem 3.15]) Let $(X,\tau)$ be a topological space without isolated
points and $\mathcal{I}$ an ideal of subsets of $X$. Then the topology
$\tau^{*}(\mathcal{I})$ is an admissible extension of $\tau$ if and only if
$\mathcal{I}\subseteq\mathcal{I}_{n}(\tau),$ where $\mathcal{I}_{n}(\tau)$ is
the family of nowhere dense sets of the space $(X,\tau)$.

In particular, $\tau^{*}(\mathcal{I}_{n}(\tau))$ is the finest admissible
extension of $\tau$ formed from $\tau$ and ideals of sets. $\Box$

Applying Theorem 5.1 and Proposition 4.2(a) we get

###### Corollary 5.1

Let $(X,\tau)$ be a topological space without isolated points and
$\mathcal{I}$, $\mathcal{J}$ be ideals of subsets of $X$ such that
$\mathcal{I}\subseteq\mathcal{J}\subseteq\mathcal{I}_{n}(\tau)$. Then
$\tau^{*}(\mathcal{J})$ is an admissible extension of $\tau^{*}(\mathcal{I})$.
$\Box$

###### Remark 5.1

The condition on the space $(X,\tau)$ ”to be without isolated points” in
Theorem 5.1 is essential. Indeed, let $(X,\tau_{\mbox{dis}})$ be a discrete
space and $\mathcal{I}=\mathcal{P}(X)$. Note that
$\tau_{\mbox{dis}}^{*}(\mathcal{I})=\tau_{\mbox{dis}}$, in particular,
$\tau_{\mbox{dis}}^{*}(\mathcal{I})$ is an admissible extension of
$\tau_{\mbox{dis}}$, and $\mathcal{I}_{n}(\tau_{\mbox{dis}})=\\{\emptyset\\}$.

###### Example 5.1

(cf. [JH]) Let $(X,\tau)$ be a topological space. Then for each $A\subseteq X$
we have $A^{*}(\mathcal{I}_{n}(\tau),\tau)=\mbox{{\rm
Cl}}_{\tau}(Int_{\tau}(\mbox{{\rm Cl}}_{\tau}(A)))$ and $\mbox{{\rm
Cl}}^{*}(A)=A\cup\mbox{{\rm Cl}}_{\tau}(Int_{\tau}(\mbox{{\rm
Cl}}_{\tau}(A)))$. Moreover, $\tau^{*}(\mathcal{I}_{n}(\tau))=\tau^{\alpha}$,
where $\tau^{\alpha}$ is a topology from [N].

Some facts about admissible extensions of the Euclidean topology $\tau_{E}$ on
the real line via ideals of sets see below.

Since the real line $(\mathbb{R},\tau_{E})$ is a Baire space we get the
following.

###### Proposition 5.1

Let $\mathcal{I}$ be an ideal of sets on the real line $(\mathbb{R},\tau_{E})$
and $\mathcal{I}\subseteq\mathcal{I}_{n}(\tau_{E})$. Then the space
$(\mathbb{R},\tau_{E}^{*}(\mathcal{I}))$ is connected. $\Box$

Let $\mathcal{T}_{\mbox{cd}}$ be the ideal of all closed discrete subsets of
the real line $(\mathbb{R},\tau_{E})$. Recall ([CN1, Proposition 4.2]) that
$\tau_{E}=\tau_{E}^{*}(\mathcal{I})$ for any ideal
$\mathcal{I}\subseteq\mathcal{I}_{\mbox{cd}}$.

###### Proposition 5.2

Let $\mathcal{I}$ be an ideal of sets on the real line $(\mathbb{R},\tau_{E})$
and $\mathcal{I}\subseteq\mathcal{I}_{n}(\tau_{E})$. Then the space
$(\mathbb{R},\tau_{E}^{*}(\mathcal{I}))$ is non-regular iff there exists an
element $I\in\mathcal{I}$ with a limit point in the topology $\tau_{E}$.
$\Box$

###### Remark 5.2

(cf. [JH]) Let $A=\\{\frac{1}{n}:n=1,2,\dots\\}\subset\mathbb{R}$ and
$\mathcal{I}_{A}$ be the ideal of all subsets of $A$. Then the topology
$\tau_{E}^{*}(\mathcal{I}_{A})$ is the nonregular Smirnov topology on
$\mathbb{R}$ from Example 4.2.

###  5.2 Hattori spaces

Other extensions of the Euclidean topology $\tau_{E}$ on the real line was
suggested in [H].

###### Definition 5.1

([H]) Let $A\subseteq\mathbb{R}$. Define a topology $\tau(A)$ on $\mathbb{R}$
as follows:

  * (1)

for each $x\in A$, $\\{(x-\epsilon,x+\epsilon):\epsilon>0\\}$ is a nbd open
basis at $x$,

  * (2)

for each $x\in\mathbb{R}\setminus A$, $\\{[x,x+\epsilon):\epsilon>0\\}$ is a
nbd open basis at $x$.

Note that $\tau(\emptyset)$ (respectively, $\tau(\mathbb{R})$) is the
Sorgenfrey topology $\tau_{S}$ (respectively, the Euclidean topology
$\tau_{E}$) on the reals, and all topologies $\tau(A),A\subseteq\mathbb{R},$
are addmissible extensions of $\tau_{E}$.

It is easy to see that for any $A,B\subseteq\mathbb{R}$ we have $A\supseteq B$
iff $\tau(A)\subseteq\tau(B)$.

So by Proposition 4.2 we get

###### Proposition 5.3

$\tau(B)$ is an admissible extension of $\tau(A)$ iff $A\supseteq B$. $\Box$

The topological spaces $H(A)=(\mathbb{R},\tau(A)),A\subseteq\mathbb{R},$ are
called Hattori spaces.

Let us recall ([CH]) that all $H(A)$-spaces are $T_{1}$ regular, hereditary
Lindelöf and hereditary separable. More on the properties of $H(A)$-spaces and
their variety see for example in [K] and [BS].

The following is evident.

###### Proposition 5.4

Any $H(A)$-space with $A\neq\mathbb{R}$ is disconnected. $\Box$

###  5.3 Almost topological groups

A paratopological group $(G,\tau)$ consists of a group $G$ and a topology
$\tau$ on $G$ that makes the group operation continuous. If in addition the
inverse operation of $G$ is continuous then $(G,\tau)$ is a topological group.

A semitopological group is a group endowed with a topology that makes
continuous left and right translations. Each paratopological group is a
semitopological one.

###### Definition 5.2

([F, Definition 2.1]) An almost topological group is a paracompact group
$(G,\tau)$ that satisfies the following conditions:

  * (a)

the group $G$ admits a Hausdorff topological group topology $\gamma$ weaker
than $\tau$, and

  * (b)

there exists a local base $\beta_{e}$ at the identity $e$ of the
paratopological group $(G,\tau)$ such that the set $U\setminus\\{e\\}$ is open
in $(G,\gamma)$ for every $U\in\beta_{e}$.

One says also that $G$ is an almost topological group with structure
$(\tau,\gamma,\beta_{e})$.

It is easy to see that the topology $\tau$ is an admissible extension of
$\gamma$.

###### Proposition 5.5

([F, Proposition 2.5]) Let $G$ be a non-discrete almost topological group with
structure $(\tau,\gamma,\beta_{e})$. Then $\gamma$ is the finest Hausdorff
topological group topology on $G$ weaker than $\tau.$

###### Remark 5.3

(cf. [CS]) For the paratopological group $(\mathbb{R},\tau_{S})$ the Euclidean
topology $\tau_{E}$ is the finest Hausdorff topological group topology on
$\mathbb{R}$ weaker than $\tau_{S}.$

###### Remark 5.4

  * (a)

The Hattori space $H(A)$, where $A$ is a proper subset of $\mathbb{R}$, is no
semitopological group.

In fact, let $x\in\mathbb{R}\setminus A$ and $y\in A$. Note that $y+(x-y)=x$
and the translation of the additive group $\mathbb{R}$ by $x-y$ is not
continuous at $y$.

  * (b)

However, the space $(\mathbb{R},\tau_{E}^{*}(\mathcal{I}_{n}(\tau_{E})))$ is a
semitopological group with open shifts and continuous inversion but it is not
a paratopological group.

In fact, since the nowhere dense sets are invariant under translations and
inversion as well as the open intervals we have that
$(\mathbb{R},\tau_{E}^{*}(\mathcal{I}_{n}(\tau_{E})))$ is a semitopological
group with open shifts and continuous inversion.

Let us show that $(\mathbb{R},\tau_{E}^{*}(\mathcal{I}_{n}(\tau_{E})))$ is no
paratopological group. Note that $0+0=0$ and
$W=(-1,1)\setminus\\{\frac{1}{n}:n=1,2,\dots\\}$ is an open neighborhood of
$0$ in $(\mathbb{R},\tau_{E}^{*}(\mathcal{I}_{n}(\tau_{E})))$. If
$(\mathbb{R},\tau_{E}^{*}(\mathcal{I}_{n}(\tau_{E})))$ is a paratopological
group then there exist two open sets $V_{1}$ and $V_{2}$ containing $0$ such
that $V_{1}+V_{2}\subseteq W$. Observe that we can find a symmetric open
neighborhood $V\subseteq V_{1}\cap V_{2}$ of $0$ and a non-empty interval
$(a,b)\subseteq V$. Since the difference $(a,b)-(a,b)$ contains a
nondegenerated interval $(-\epsilon,\epsilon)$ we have
$(-\epsilon,\epsilon)\subseteq V-V\subseteq V_{1}+V_{2}\subseteq W$. But this
is impossible.

###### Example 5.2

([F, Example 2.2]) There exist non-regular almost topological groups. In fact,
consider the additive group $\mathbb{R}^{2}$ with the identity $e=(0,0)$. For
every $r>0$ let
$B_{r}=\\{e\\}\cup\\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}<r^{2},x>0\\}$. It is
easy to see that the family $\mathcal{B}=\\{B_{r}:r>0\\}$ is a local base at
$e$ of a Hausdorff paratopological group. The group $(\mathbb{R}^{2},\tau)$,
where $\tau$ is a topology generated by the family $\mathcal{B}$, is a
Hausdorff non-regular almost topological group.

###  5.4 Hattori topologies on almost topological groups

We will follow ([CS]) and apply notations from Definition 5.2.

Let $(G,\tau)$ be an almost topological group and $\beta_{e}$ a local base at
the identity $e$ of $(G,\tau)$. Then $\\{UU^{-1}:U\in\beta_{e}\\}$ is a local
base at $e$ in the Hausdorff topological group $(G,\gamma)$. Given $x\in G$,
let $\beta_{x}=\\{Ux:U\in\beta_{e}\\}$ and
$\beta_{e}^{\prime}=\\{UU^{-1}x:U\in\beta_{e}\\}$.

Given $A\subseteq G$, consider the collection $\\{\beta(x):x\in G\\}$, where

$\beta(x)=\begin{array}[]{l}\beta_{x}^{\prime}.\ \mbox{if}\ x\in A,\\\
\beta_{x}.\ \mbox{if}\ x\notin A.\end{array}$

###### Theorem 5.2

([CS, Theorem 3.1]) Let $(G,\tau)$ be an almost topological group and
$A\subseteq G$. Then $\\{\beta(x):x\in G\\}$ is a neighborhood system for a
topology $\tau(A)$ on $G$. $\Box$

The topological space $(G,\tau(A))$ is denoted by $H(A,G)$ and it is called
the Hattori space of $G$ associated to $A$. Let us note that
$H(\emptyset,G)=(G,\tau)$ and $H(G,G)=(G,\gamma)$.

###### Proposition 5.6

([CS, Proposition 3.6]) Let $(G,\tau)$ be an almost topological group such
that $(G,\tau)$ is not a topological group. If $A,B\subseteq G$ then
$\tau(A)\subseteq\tau(B)$ iff $B\subseteq A$. $\Box$

Proposition 5.6 and Proposition 4.2 imply

###### Proposition 5.7

Let $(G,\tau)$ be an almost topological group such that $G$ is not a
topological group, and $A,B\subseteq G$. Then $\tau(A)$ is an admissible
extension of $\tau(B)$ iff $A\subseteq B$. $\Box$

###### Problem 5.1

Let $(G,\tau)$ be the almost topological group and $\gamma$ be the finest
Hausdorff topological group topology on $G$ weaker than $\tau.$ (For example,
let $(G,\tau)$ be the almost topological group from Example 5.2.) Describe the
topological diversity of Hattori spaces $H(A,G),A\subseteq G$.

## References

  * [BS] A. Bouziad, E. Sukhacheva, On Hattori spaces, Comment. Math. Univ. Carolin. 58,2 (2017) 213-223
  * [CS] A. Calderon-Villabos, I. Sanchez, Hattori topologies on almost topological groups, Topology Appl. 326 (2023), Paper No. 108411, 15 pp
  * [CH] V. A. Chatyrko, Y. Hattori, A poset of topologies on the set of real numbers, Comment. Math. Univ. Carolin. 54, 2 (2013) 189-196.
  * [CN1] V. A. Chatyrko, V. Nyagahakwa, Sets with the Baire property in topologies formed from a given topology and ideals of sets, Questions and answers in General Topology, 35 (2017) 59-76
  * [CN2] V. A. Chatyrko, V. Nyagahakwa, On the families of sets without the Baire property generated by Vitali sets, P-Adic Numbers, Ultrametric Analysis, and Applications, 3 (2011), 100–107
  * [E] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
  * [F] M. Fernandez, On some classes of paratopological groups, Topol. Proc. 40 (2012) 63-72
  * [H] Y. Hattori, Order and topological structures of poset of the formal balls on metric spaces, Mem.Fac.Sci.Eng. Shimane Univ. Ser. B Math. Sci., 43 (2010) 13 - 26.
  * [JH] D. Jankovic, T. R. Hamlett, New topologies from old via Ideals, Amer. Math. Monthly vol. 97 (1990) No. 4, 295-310.
  * [Kh] A. B. Kharazishvili, Nonmeasurable Sets and Functions, Elsevier, Amsterdam, 2004.
  * [K] J. Kulesza, Results on spaces between the Sorgenfrey topology and the usual topology on $\mathbb{R}$, Topol. Appl. 231 (2017) 266-275
  * [Ku] K. Kuratowski, Topology, vol. 1, Academic Press, New York and London, 1966
  * [N] O. Njåstad, On some classes of nearly open sets, Pacific J. Math. 15 (1965) p. 961-970
  * [SS] L. A. Steen, J. A. Seebach, Jr. Counterexamples in topology, Springer-Verlag, New York, 1978

(V.A. Chatyrko)
Department of Mathematics, Linkoping University, 581 83 Linkoping, Sweden.
<EMAIL_ADDRESS>


# Iterating skew evolutes and skew involutes: a linear analog of the bicycle
kinematics

Serge Tabachnikov111 Department of Mathematics, Pennsylvania State University,
University Park, PA 16802, USA<EMAIL_ADDRESS>
###### Abstract

The evolute of a plane curve is the envelope of its normals. Replacing the
normals by the lines that make a fixed angle with the curve yields a new
curve, called the evolutoid. We prefer the term “skew evolute”, and we study
the geometry and dynamics of the skew evolute map and of its inverse, the skew
involute map. The relation between a curve and its skew evolute is analogous
to the relation between the rear and front bicycle tracks, and this
connections with the bicycle kinematics (a considerably more complicated
subject) is our motivation for this study.

## Introduction and motivation.

The evolute of a smooth plane curve is the envelope of its normals. In this
article we consider the following modification of this construction.

Figure 1: A skew evolute of an ellipse.

Let ${\gamma}$ be a smooth oriented curve and $\alpha$ a fixed angle. Turn
each tangent line of ${\gamma}$ through angle $\alpha$ about the tangency
point, and let ${\Gamma}$ be the envelope of this 1-parameter family of lines.
We call ${\Gamma}$ a skew evolute of ${\gamma}$ and write
${\Gamma}={\mathcal{E}}_{\alpha}({\gamma})$. See Figure 1. The usual evolute
corresponds to the case $\alpha=\pi/2$, and if $\alpha=0$, then
${\Gamma}={\gamma}$. Likewise, we call ${\gamma}$ a skew involute of
${\Gamma}$ and write ${\gamma}={\mathcal{I}}_{\alpha}({\Gamma})$.

This subject goes back to the early 18th century, see [13] (we learned about
this reference from [6]). However, it continues to attract attention; see [1,
2, 6, 8, 10, 14, 21] for a sampler of this century work.

What we called “skew evolute” is traditionally called “evolutoid”. The reason
we use a different term is to emphasize the similarity with the classical
evolutes and involutes. Indeed, what we call “skew involutes” were called
“tanvolutes” in [2]. It seems that the terminology has not completely
crystalized yet.

A study of skew evolutes necessarily involves curves with cusps; indeed, the
evolute of a closed simple curve has at least four cusps, as the classical
4-vertex theorem implies. We study the dynamics of the transformations
${\mathcal{E}}_{\alpha}$ and ${\mathcal{I}}_{\alpha}$ on the class of curves
called hedgehogs.

An oriented line in the plane is characterized by its direction $\alpha$ and
the support number $p$, the signed distance from the origin to the line,
Figure 2. The coorientation of an oriented line is given by the direction
$\phi=\alpha-\pi/2$.

Figure 2: The space of (co)oriented lines.

Let ${\gamma}$ be an oriented smooth strictly convex closed curve. It can be
parameterized by $\phi\in S^{1}={\mathbb{R}}/2\pi{\mathbb{Z}}$, the direction
of its outward normal vectors, and the support numbers of the tangent lines
are given by a function $p(\phi)$. This is the support function of ${\gamma}$.
The support function uniquely characterizes the curve and is determined by the
curve, except that a change of the origin amounts to adding to $p(\phi)$ a
first harmonic, a linear combination of $\cos\phi$ and $\sin\phi$.

In particular, the equation of the curve, defined by its support function, is

${\gamma}(\phi)=(p(\phi)\cos\phi-p^{\prime}(\phi)\sin\phi,p(\phi)\sin\phi+p^{\prime}(\phi)\cos\phi).$

The length of ${\gamma}$ and the area bounded by it are given by

$L=\int_{0}^{2\pi}p(\phi)\ d\phi,\ \
A=\frac{1}{2}\int_{0}^{2\pi}[p^{2}(\phi)-(p^{\prime})^{2}(\phi)]\ d\phi,$

and the curvature radius of ${\gamma}$ by $p(\phi)+p^{\prime\prime}(\phi)$,
see, e.g., [15].

Replacing a curve by its equidistant curve amounts to adding a constant to the
support function. The curves, that are equidistant to convex ones, still do
not have inflections and are characterized by their support functions
$p:S^{1}\to{\mathbb{R}}$, but they may have cusps, where the radius of
curvature vanishes. The tangent lines are well defined at the cusps, and the
coorientation is continuous therein (unlike the orientation that is reversed
at the cusps).

The cooriented curves described by the support functions are called hedgehogs.
The orientation of a smooth arc of a hedgehog is obtained from its
coorientation by a $90^{\circ}$ rotation in the positive direction. An
equivalent characterization of hedgehogs is that they are equidistant to
convex curves. The above formulas for the length and area are still valid, but
these quantities are signed (for example, the sign of the length changes as
one traverses a cusp).

A hypocycloid is a hedgehog whose support function is a pure harmonic, a
linear combination of $\cos(k\phi)$ and $\sin(k\phi)$. The number $k\geq 2$ is
the order of a hypocycloid, see Figure 3. We consider circles as the
hypocycloids of order zero.

Figure 3: Hypocycloids of order $2,3$, and $4$ (the middle curve is traversed
twice).

There are two motivations for this study. First, this is a generalization of
the work done in [3], where iterations of evolutes and involutes were
considered, both in the continuous and discrete settings (when curves are
replaced by polygons).

Second, there is a relation with the recent study of bicycle kinematics that
we now describe.

Bicycle is modeled as an oriented segment of fixed length $L$ that can move in
such a way that the velocity of its rear end is always aligned with the
segment (the rear wheel is fixed on the bicycle frame). The bicycle leaves two
tracks, the rear track ${\gamma}$ and the front track ${\Gamma}$, and they are
related as shown on the left of Figure 4. See, e.g., [4, 5, 11].

Figure 4: The correspondence between the rear and front bicycle tracks and the
dual picture.

This model of bicycle can be also considered in the spherical geometry, see
[9]. In the spherical geometry one has a duality between points and oriented
great circles, the pole-equator correspondence. This spherical duality extends
to smooth curves and, applied to the left part of Figure 4, it yields the
right part, where the angle $\alpha$ equals the spherical length $L$.

However, we consider the right part of Figure 4 as drawn in the plane. In this
way, the map that takes the rear bicycle track to the front track is analogous
to the map that takes a curve to its skew evolute. Unlike the former map, the
latter one is linear, and it is much easier to study. In particular, as we
shall see, the bicycle analogs of some results concerning skew evolutes and
skew involutes remain open problems.

## Known results.

In this section we present known results on skew evolutes and skew involutes.

Let ${\gamma}$ be a hedgehog with the support function $p(\phi)$, and let
${\Gamma}={\mathcal{E}}_{\alpha}$ be its skew-evolute with the support
function $q(\phi)$. Then

$q(\phi)=p(\phi-\alpha)\cos\alpha+p^{\prime}(\phi-\alpha)\sin\alpha,$ (1)

see [10]. In particular, $L({\Gamma})=\cos\alpha\ L({\gamma})$.

Denote the linear differential operator on the right hand side of (1) by
${\mathcal{D}_{\alpha}}(p)(\phi)$.

The Steiner point, or the curvature centroid, $St({\gamma})$, of a curve
${\gamma}$ is its center of mass with the density equal to the curvature. In
terms of the support function, it is given by

$St({\gamma})=\frac{1}{\pi}\int_{0}^{2\pi}p(\phi)\ (\cos\phi,\sin\phi)\
d\phi.$

A hedgehog ${\gamma}$ and its skew evolute
${\Gamma}={\mathcal{E}}_{\alpha}({\gamma})$ share their Steiner points, see
[1].

For a quick proof, note that the Steiner point is characterized by the
condition that, if it is chosen as the origin, then the support function is
$L^{2}$-orthogonal to the first harmonics, that is, its Fourier expansion does
not contain the first harmonics. This property is preserved by the operator
${\mathcal{D}}_{\alpha}$, and the result follows.

The evolute of a curve is the locus of the centers of its osculating circles.
For skew evolutes, the role of circles is played by the logarithmic spirals.

A logarithmic spiral centered at the origin is characterized by the property
that the position vector of every point makes a constant angle $\alpha$ with
the direction of the curve at this point. If $\alpha=\pi/2$, the spiral is a
circle.

Call such logarithmic spirals $\alpha$-spirals. They form a 1-parameter family
of curves. Allowing parallel translation of the origin, results in a
3-parameter family of $\alpha$-spiral (similarly to circles). It follows that,
for every $\alpha$, a smooth curve has an osculating $\alpha$-spiral at every
point (it approximates the curve to second order). A hyper-osculating
$\alpha$-spiral is tangent to the curve to higher order.

Therefore the skew evolute ${\mathcal{E}}_{\alpha}({\gamma})$ is the locus of
centers of the osculating $\alpha$-spirals of the curve ${\gamma}$. The cusps
of ${\mathcal{E}}_{\alpha}({\gamma})$ correspond to hyper-osculating
$\alpha$-spirals, see [20].

The cusps of a skew evolute happen when its radius of curvature $r(\phi)$
vanishes. In view of equation (1), this amounts to the equation
$r\cos\alpha+r^{\prime}\sin\alpha=0$, or $r^{\prime}/r=-\cot\alpha$. See [6]
for a study of cusps of skew evolutes.

Let ${\Gamma}$ be a hedgehog. Given $\alpha$, does ${\Gamma}$ have a closed
skew-involute, and if so, how many? For $\alpha=\pi/2$, the involute is
provided by the string construction, and a necessary and sufficient condition
for it to close up is that the signed length of ${\Gamma}$ vanishes, in which
case one has a 1-parameter family of involutes.

However, if $\alpha\neq\pi/2$, then there exists a unique closed skew-involute
${\mathcal{I}}_{\alpha}({\Gamma})$, see [2]. The reason is that the monodromy
of the linear differential equation (1) is a homothety of the real line with
coefficient $\neq 1$. Such a map has a unique fixed point, corresponding to
the desired periodic solution.

Comparing with the bicycle kinematics, we observe the following difference.
Given a closed front bicycle track, the rear track is determined by a first
order ordinary Riccati differential equation, see, e.g., [5]. Unlike the case
of skew involutes, the monodromy of this equation takes values in the group
$SL(2,{\mathbb{R}})$, acting on the circle ${\mathbb{RP}}^{1}$ of the initial
positions of the bicycle by fractional-linear transformations. Generically,
such a transformation has none or two fixed points.

## Three examples.

The following examples concern locally convex curves that are not closed, and
their support functions are not periodic anymore. However formula (1) is still
valid.

Cycloid. It is well known that the evolute of a cycloid is congruent to the
cycloid by parallel translation. The same holds for skew evolutes, see Figure
5.

Indeed, the support function of a cycloid is $p(\phi)=-\phi\cos\phi$. Using
equation (1), we find that the support function of the skew evolute is

$q(\phi)=-\phi\cos\phi+(\alpha-\cos\alpha\sin\alpha)\cos\phi+\sin^{2}\alpha\sin\phi.$

Thus the support function has changed by a first harmonic, which amounts to a
parallel translation of the curve.

Figure 5: Left: a cycloid and its skew evolute. Right: a logarithmic spiral
and its skew evolute.

Logarithmic spiral. Logarithmic spirals are congruent to their skew evolutes
by rotation, see Figure 5. Indeed, the support function of a logarithmic
spiral is $p(\phi)=e^{c\phi}$. Hence the support function of its skew evolute
is

$q(\phi)=e^{-c\alpha}(\cos\alpha+c\sin\alpha)e^{c\phi},$

which is obtained from $e^{c\phi}$ by a parameter shift. If $c=-\cot\alpha$,
the skew evolute reduces to a point.

A slight generalization is a curve ${\gamma}$ whose support function is
$p(\phi)=c_{1}e^{b_{1}\phi}+c_{2}e^{b_{2}\phi}$. If

$(\cos\alpha+b_{1}\sin\alpha)^{b_{2}}=(\cos\alpha+b_{2}\sin\alpha)^{b_{1}},$

then ${\mathcal{E}}_{\alpha}({\gamma})$ is congruent to ${\gamma}$ by
rotation.

Parabola. A calculation, that we do not present, shows that the skew evolute
of the parabola $(t,t^{2}/2)$ has a cusp for $3t=-\cot\alpha$, see Figure 6.
Thus the skew evolute of a parabola has a unique cusp for every
$\alpha\in(0,\pi)$.

Figure 6: Skew evolutes of a parabola with $\alpha=\pi/10,\pi/5,3\pi/10$, and
$2\pi/5$.

## New results.

Now we present results that, to the best of our knowledge, are not found in
the literature.

First, let us look at the above defined linear operator
${\mathcal{D}_{\alpha}}$ in detail. It preserves the 2-dimensional space of
$k$th harmonics. In the basis $(\cos(k\phi),\sin(k\phi))$, it is given by the
matrix

$\begin{pmatrix}\cos^{2}\alpha+k\sin^{2}\alpha,&(k-1)\cos\alpha\sin\alpha\\\
-(k-1)\cos\alpha\sin\alpha,&\cos^{2}\alpha+k\sin^{2}\alpha\end{pmatrix}.$ (2)

This is a linear similarity, a composition of rotation and dilation; the
dilation coefficient is equal to $\sqrt{1+(k^{2}-1)\sin^{2}\alpha}$. In
particular, for $\alpha\neq\pi/2$, the operator ${\mathcal{D}_{\alpha}}$ is
invertible.

Since similarities with a fixed center commute, we have

###### Corollary 1

One has

${\mathcal{E}}_{\alpha}\circ{\mathcal{E}}_{\beta}={\mathcal{E}}_{\beta}\circ{\mathcal{E}}_{\alpha}\
\ {\rm and}\ \
{\mathcal{I}}_{\alpha}\circ{\mathcal{I}}_{\beta}={\mathcal{I}}_{\beta}\circ{\mathcal{I}}_{\alpha}.$

Next we ask how the shape of a hedgehog evolves under iterations of the skew
evolute or skew involute operations.

###### Theorem 1

(i) Assume that the support function of ${\gamma}$ is a trigonometric
polynomial of degree $d$. Then the iterated skew evolutes of ${\gamma}$
converge, in shape, to a hypocycloid of order $d$.
(ii) If the Fourier series of the support function of ${\gamma}$ has a free
term, then its iterated skew involutes converge, in shape, to a circle. If the
Fourier series starts with $d$th harmonics, then the iterated skew involutes
converge, in shape, to a hypocycloid of order $d$.

## Proof.

For the first statement, formula (2) implies that, under iterations, the
highest harmonics grow faster than the lower ones. This implies the result.

Likewise, under ${\mathcal{D}_{\alpha}}^{-1}$, the free term of the Fourier
series is multiplied by $1/\cos\alpha>1$, whereas the space of $k$th harmonics
is stretched by the factor $1/\sqrt{1+(k^{2}-1)\sin^{2}\alpha}<1$. In the
first case, the free term dominates under iterations, and in the second case,
so does the first non-trivial harmonic. $\Box$

###### Corollary 2

A hedgehog is similar to its skew involute if and only if it is a hypocycloid.

In the case of evolutes and involutes ($\alpha=\pi/2$), the above results were
obtained in [3]. The next theorem extends another result in [3] from evolutes
to skew evolutes.

###### Theorem 2

Assume that the support function $p(\phi)$ of a hedgehog ${\gamma}$ is not a
trigonometric polynomial, that is, its Fourier expansion contains infinitely
many terms. Then the number of cusps of the iterated skew evolutes increases
without bound.

## Proof.

The proof consists of two steps.

Claim 1: The number of sign changes of the functions
${\mathcal{D}}_{\alpha}^{n}(p)$ increases without bound as $n\to\infty$.

This is a slight generalization of the theorem by Polya and Wiener [12] where
the case of the operator $p\mapsto p^{\prime}$ is considered. Since this
argument is not sufficiently well known, we present it here.

Let

$p(\phi)=\sum_{k\in{\mathbb{Z}}}a_{k}e^{ki\phi},\ a_{-k}=\bar{a}_{k},$

be the Fourier expansion of $p$. It suffices to proof the statement for a
simpler operator ${\mathcal{F}}(p)=p^{\prime}+cp$, that is,

${\mathcal{F}}:p\mapsto\sum_{k\in{\mathbb{Z}}}(c+ik)a_{k}e^{ki\phi}.$

The claim is that if $a_{m}\neq 0$ then, for sufficiently large $n$, the
function ${\mathcal{F}}^{n}(p)$ has at least $2m$ sign changes.

Let ${\mathcal{Z}}(f)$ denote the number of sign changes of a periodic
function $f$. A version of Rolle’s theorem, Lemma 1 in [12], asserts that, for
every $b\in{\mathbb{R}}$,

${\mathcal{Z}}\left(\sum_{k\in{\mathbb{Z}}}a_{k}e^{ki\phi}\right)\geq{\mathcal{Z}}\left(\sum_{k\in{\mathbb{Z}}}\frac{a_{k}e^{ki\phi}}{b^{2}+k^{2}}\right).$

Apply this to $f={\mathcal{F}}^{n}(p)$, $b^{2}=m^{2}+2c^{2}$, and iterate the
inequality $n$ times:

$\begin{split}{\mathcal{Z}}\left(\sum_{k\in{\mathbb{Z}}}(c+ik)^{n}a_{k}e^{ki\phi}\right)\geq&{\mathcal{Z}}\left(\sum_{k\in{\mathbb{Z}}}\frac{(c+ik)^{n}a_{k}e^{ki\phi}}{(m^{2}+2c^{2}+k^{2})^{n}}\right)=\\\
&{\mathcal{Z}}\left(\sum_{k\in{\mathbb{Z}}}\frac{2[\sqrt{c^{2}+m^{2}}(c+ik)]^{n}a_{k}e^{ki\phi}}{(m^{2}+2c^{2}+k^{2})^{n}}\right).\end{split}$

Let $q_{n}(\phi)=\sum_{k\in{\mathbb{Z}}}c_{k}e^{ki\phi}$ be the function on
the right. Then

$|c_{k}|=\left(\frac{2\sqrt{c^{2}+m^{2}}\sqrt{c^{2}+k^{2}}}{m^{2}+2c^{2}+k^{2}}\right)^{n}|a_{k}|.$

One has

$\frac{2\sqrt{c^{2}+m^{2}}\sqrt{c^{2}+k^{2}}}{m^{2}+2c^{2}+k^{2}}<1,$

unless $k=m$, in which case this coefficient equals 1. This implies that, for
sufficiently large $n$,

$|c_{m}|>\sum_{k\neq m}|c_{k}|,$

as in [12]. For such $n$, ${\mathcal{Z}}(q_{n})$ equals the number of sign
changes of its $m$th harmonic, that is, equals $2m$, as needed.

Claim 2: If the support function of a hedgehog ${\gamma}$ has $2m$ sign
changes, then ${\gamma}$ has at least $m$ cusps.

Indeed, if the support function of ${\gamma}$ has $2m$ zeros, then there are
$2m$ tangents from the origin $O$ to ${\gamma}$. Each arc of a hedgehog
between its cusps is convex, and there are at most two tangents from $O$ to
it. Therefore there must be at least $m$ such arcs, and at least as many
cusps. $\Box$

Theorems 1 and 2 imply

###### Corollary 3

If all iterated skew evolutes of a hedgehog ${\gamma}$ are free from cusps,
then ${\gamma}$ is a circle.

What is an analog of this statement in terms of the bicycle model? Since the
projective duality interchanges cusps and inflections, we are led to the
following formulation.

Let ${\gamma}$ be a smooth oriented closed curve, $L$ a fixed positive number.
Denote by ${\mathcal{T}}({\gamma})$ the locus of endpoints of the positve
tangent segments to ${\gamma}$ of length $L$. That is,
${\mathcal{T}}({\gamma})$ is the front track of the bicycle whose rear track
is ${\gamma}$.

###### Conjecture 4

Assume that all iterations ${\mathcal{T}}^{k}({\gamma}),k\geq 0,$ are convex
curves. Then ${\gamma}$ is a circle.

## An “integrable” map on hedgehogs.

Given a bicycle rear track, one can traverse it in the opposite directions,
creating two front tracks. This relation between curves is completely
integrable, see [4]. Equivalently, two smooth curves, ${\Gamma}_{1}$ and
${\Gamma}_{2}$, are in the bicycle correspondence if two points, $x_{1}$ and
$x_{2}$, can traverse them in such a way that the distance between them
remains constant (twice the bicycle frame) and the velocity of the midpoint of
the segment $x_{1}x_{2}$ is aligned with this segment (that is how the read
wheel moves).

An analog of this relation in our setting is as follows.

Fix an angle $\alpha$ and consider a hedgehog ${\Gamma}_{1}$. One constructs
its skew-involute ${\gamma}$, and then ${\Gamma}_{2}$, the skew-evolute of
${\gamma}$ with the angle $-\alpha$. We obtain a map
${\mathcal{M}_{\alpha}}={\mathcal{E}}_{-\alpha}\circ{\mathcal{I}}_{\alpha}:{\Gamma}_{1}\mapsto{\Gamma}_{2}$.
See Figure 7.

Figure 7: Left: the image of an ellipse under the map
${\mathcal{M}_{\alpha}}$. Right: the curve with the support function
$p(\phi)=e^{2\sin\phi}$ (black) and its images under ${\mathcal{M}_{\alpha}}$
for $\alpha=0.5,0.9$ and $1.2$ (blue, red, and green, respectively).

Equivalently, two points, $x_{1}$ and $x_{2}$, traverse the curves
${\Gamma}_{1}$ and ${\Gamma}_{2}$ in such a way that the angle between the
(co)oriented tangent lines at $x_{1}$ and $x_{2}$ is $2\alpha$, and the
intersection point of these lines moves in the direction of the bisector
between these oriented lines, see Figure 8.

Figure 8: The map ${\mathcal{M}_{\alpha}}$.

Let $p_{1}$ and $p_{2}$ be the support functions of these curves. The next
formula follows from equation (1):

$p_{2}^{\prime}(\phi)-p_{2}(\phi)\cot\alpha=-p_{1}^{\prime}(\phi-2\alpha)-p_{1}(\phi-2\alpha)\cot\alpha.$
(3)

The next lemma lists some properties of the maps ${\mathcal{M}_{\alpha}}$.

###### Lemma 5

1) A curve ${\Gamma}$ and ${\mathcal{M}_{\alpha}}({\Gamma})$ share their
Steiner points;
2) The maps commute:
${\mathcal{M}_{\alpha}}\circ{\mathcal{M}_{\beta}}={\mathcal{M}_{\beta}}\circ{\mathcal{M}_{\alpha}}$;
3) One has ${\mathcal{M}_{\alpha}}^{-1}={\mathcal{M}_{-\alpha}}$.

## Proof.

The first two properties follow from those of the skew evolute map. For the
third, one has ${\mathcal{I}_{\alpha}}={\mathcal{E}_{\alpha}}^{-1}$, hence
${\mathcal{M}_{\alpha}}={\mathcal{E}_{-\alpha}}\circ{\mathcal{E}_{\alpha}}^{-1}$.
It follows that
${\mathcal{M}_{\alpha}}^{-1}={\mathcal{E}_{\alpha}}\circ{\mathcal{E}_{-\alpha}}^{-1}={\mathcal{M}_{-\alpha}}$.
$\Box$

The maps ${\mathcal{M}_{\alpha}}$ are integrable in the following sense.

###### Lemma 6

For every $k$ and every $\alpha$, the sum of the squares of $k$th Fourier
coefficients of the support function is preserved by the map
${\mathcal{M}_{\alpha}}$: if

$p(\phi)=\sum_{k\in{\mathbb{Z}}}a_{k}e^{ki\phi},\ a_{-k}=\bar{a}_{k},$

is a Fourier expansion of the support function of ${\Gamma}$, then the
amplitude $|a_{k}|$ is an integral of the map ${\mathcal{M}_{\alpha}}$ for
every $k\geq 0$.

## Proof.

As before, the map preserves the the 2-dimensional spaces of $k$th harmonics.
A direct calculation, using equation (3), shows that this map is a rotation.
More precisely, define the angle $\beta_{k}$ by $\tan\beta_{k}=k\tan\alpha$.
Then the Fourier coefficients are transformed as follows:

$a_{k}\mapsto a_{k}e^{2i(\beta_{k}-\alpha)},$

and the action of ${\mathcal{M}_{\alpha}}$ on the space of $k$th harmonics is
the rotation by $2(\beta_{k}-\alpha)$. $\Box$

In particular, hypocycloids evolve by rotations. In this sense, they behave as
solitons of the map ${\mathcal{M}_{\alpha}}$.

In addition to the signed length $L({\Gamma})$ and signed area $A({\Gamma})$,
let $R({\Gamma})=\int_{0}^{2\pi}r^{2}(\phi)d\phi$, where $r(\phi)$ is the
curvature radius of the curve ${\Gamma}$.

###### Corollary 7

One has

$L({\Gamma})=L({\mathcal{M}_{\alpha}}({\Gamma})),\
A({\Gamma})=A({\mathcal{M}_{\alpha}}({\Gamma})),\ {\rm and}\
R({\Gamma})=R({\mathcal{M}_{\alpha}}({\Gamma})).$

## Proof.

The first equality directly follows from equation (3).

Let

$p(\phi)=\sum_{k\in{\mathbb{Z}}}a_{k}e^{ki\phi},\
q(\phi)=\sum_{m\in{\mathbb{Z}}}a_{m}e^{mi\phi},\ a_{-k}=\bar{a}_{k},\
b_{-m}=\bar{b}_{m},$

be the Fourier expansions of two periodic functions. Then

$\frac{1}{2\pi}\int_{0}^{2\pi}p(\phi)q(\phi)d\phi=a_{0}b_{0}+\sum_{k>0}(a_{k}\bar{b}_{k}+\bar{a}_{k}b_{k}).$

Let $p(\phi)$ be the support function of ${\Gamma}$. Then
$r(\phi)=p(\phi)+p^{\prime\prime}(\phi)$, and

$A({\Gamma})=\frac{1}{2}\int_{0}^{2\pi}(p^{2}(\phi)-p^{\prime
2}(\phi))d\phi,\quad
R({\Gamma})=\int_{0}^{2\pi}(p(\phi)+p^{\prime\prime}(\phi))^{2}d\phi,$

see, e.g, [15]. It follows that

$\pi A({\Gamma})=a_{0}^{2}+2\sum_{k>0}(1-k^{2})|a_{k}|^{2},\
R({\Gamma})=a_{0}^{2}+2\sum_{k>0}(1-k^{2})^{2}|a_{k}|^{2},$

and the result follows from Lemma 6. $\Box$

Consider the space of hedgehogs whose support functions are trigonometric
polynomials of degree $d$. This space is $2d+1$-dimensional. Fixing the
amplitudes of each harmonic, we obtain a space ${\mathcal{H}}_{d}$, a
$d$-dimensional torus. If ${\Gamma}\in{\mathcal{H}}_{d}$, then so is
${\mathcal{M}_{\alpha}}({\Gamma})$. Geometrically, this space consists of the
Minkowski sums of hypocycloids of orders $0,1,\ldots,d$, scaled according to
the fixed amplitudes, and each rotated through all angles independently of
each other.

The map ${\mathcal{M}_{\alpha}}$ is a rotation of this torus: the $k$th factor
$S^{1}$ is rotated by $2(\beta_{k}-\alpha)$, where $\beta_{k}$ are as in the
proof of Lemma 6. For a generic $\alpha$, it is natural to expect the angles
$\beta_{k}$ to be rationally independent.

###### Conjecture 8

For a generic $\alpha$, the orbit of a point is dense in the torus
${\mathcal{H}}_{d}$.

## Gutkin vs Wegner.

Circles are invariant under ${\mathcal{M}_{\alpha}}$ for every $\alpha$. Are
there other invariant curves?

This question is an analog of the following “bicycle” problem: which curves
are in the bicycle correspondence with themselves? This problem is equivalent
to Ulam’s problem to describe the bodies that float in equilibrium in all
positions (in dimension 2), problem 19 of The Scottish Book [16].

This problem is not completely solved, but there is a wealth of results in
this direction, including constructions of such curves by F. Wegner: these
curves are pressurized elastica, and they are solitons of the planar filament
equation, a completely integrable partial differential equation of soliton
type. See [17, 18, 19] and [4].

However, due to linearity, the problem at hand is considerably simpler, and it
was solved by E. Gutkin in the billiard set-up [7].

Indeed, if ${\mathcal{M}_{\alpha}}({\Gamma})={\Gamma}$ for a convex curve
${\Gamma}$, and ${\gamma}={\mathcal{I}_{\alpha}}({\Gamma})$ is also convex,
then ${\Gamma}$ is a caustic of the billiard inside ${\gamma}$, having the
special property that the billiard trajectories tangent to ${\Gamma}$ make
angle $\alpha$ with the billiard curve ${\gamma}$; see also [1].

######  Theorem 3 (Gutkin)

A necessary and sufficient condition for such non-circular curves ${\Gamma}$
to exist is that $k\tan\alpha=\tan(k\alpha)$ for some $k\geq 2$.

## Proof.

To show necessity, set $p_{2}=p_{1}=:p$ in (3) and rewrite it as

$p_{2}^{\prime}(\phi+\alpha)\sin\alpha-
p_{2}(\phi+\alpha)\cos\alpha+p_{1}(\phi-\alpha)\cos\alpha+p_{1}^{\prime}(\phi-\alpha)\sin\alpha=0.$
(4)

If

$p(\phi)=p_{0}+\sum_{1}^{\infty}a_{k}\cos(k\phi)+b_{k}\sin(k\phi),$

then equation (4) implies

$a_{k}(\sin(k\alpha)\cos\alpha-k\cos(k\alpha)\sin\alpha)=b_{k}(\sin(k\alpha)\cos\alpha-k\cos(k\alpha)\sin\alpha)=0.$

If the curve is not a circle, then $a_{k}\neq 0$ or $b_{k}\neq 0$ for some
$k\geq 2$, and then

$\sin(k\alpha)\cos\alpha=k\cos(k\alpha)\sin\alpha,$

as needed.

For sufficiency, one can take a “fattened” hypocycloid of order $k$, that is,
add a sufficiently large constant to the support function of the hypocycloid.
This yields a convex curve having the desired property. $\Box$

Acknowledgements. I am grateful to M. Arnold, D. Fuchs, A. Glutsyuk, J.
Jerónimo-Castro, and I. Izmestiev for discussions, suggestions, and help. I
was supported by NSF grant DMS-2005444.

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