# Supplemental Material for “A physically motivated analytical expression for
the temperature dependence of the zero-field splitting of the nitrogen-vacancy
center in diamond”

M. C. Cambria  Department of Physics, University of Wisconsin, Madison, WI
53706, USA  G. Thiering  Wigner Research Centre for Physics, P.O. Box 49, 1525
Budapest, Hungary  A. Norambuena  Centro de Optica e Información Cuántica,
Universidad Mayor, Camino la Piramide 5750, Huechuraba, Santiago, Chile  H. T.
Dinani  Escuela de Ingeniería, Facultad de Ciencias, Ingeniería y Tecnología,
Universidad Mayor, Santiago, Chile  A. Gardill  I. Kemeny  Department of
Physics, University of Wisconsin, Madison, WI 53706, USA  V. Lordi  Lawrence
Livermore National Laboratory, Livermore, CA, 94551, USA  Á. Gali  Wigner
Research Centre for Physics, P.O. Box 49, 1525 Budapest, Hungary  Department
of Atomic Physics, Institute of Physics, Budapest University of Technology and
Economics, Műegyetem rakpart 3., 1111 Budapest, Hungary  J. R. Maze  Instituto
de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago,
Chile  Centro de Investigación en Nanotecnología y Materiales Avanzados,
Pontificia Universidad Católica de Chile, Santiago, Chile  S. Kolkowitz
<EMAIL_ADDRESS>Department of Physics, University of Wisconsin,
Madison, WI 53706, USA  Department of Physics, University of California,
Berkeley, CA 94720, USA

††preprint: APS/123-QED

##  I Experimental details

The experiments in this work were conducted using a modified version of the
apparatus described previously in Ref. [1]. In this section we will describe
the main features of the apparatus and how it differs from the one used in
Ref. [1]. The apparatus consists of a home-built confocal microscope with low-
and high-temperature operation modes. In low-temperature operation mode
(15-295 K), temperature control is provided by an attocube attoDRY800 closed-
cycle cryostat. In order to minimize the gradient between the temperature
sensor and the diamond sample, an additional sensor (Lakeshore Cernox
CX-1050-SD-HT-1.4L-Q) was installed on top of the cold finger immediately
adjacent to the sample. The additional temperature sensor was read out via a
4-wire measurement performed with a Lakeshore 218 monitor. The sensor and
diamond sample were adhered to the sample using Ted Pella Leitsilber 200
silver paste. We note that a gradual buildup of ice on the diamond sample
below 70 K led to increased optical background and resulted in the
comparatively large error bars apparent in measurements near 50 K. In high-
temperature operation mode (296-500 K), temperature control is provided by a
homebuilt copper hot plate with custom PID control software. The hot plate
temperature sensor (TE Connectivity PT100 NB-PTCO-024) was positioned
immediately adjacent to the sample and read out via a 4-wire measurement
performed with a multimeter (Multicomp Pro MP730028). The temperature sensor
and diamond sample were adhered to the hot plate using Ted Pella PELCO High
Performance Nickel Paste. In both operation modes, temperature readings were
logged once every 2 seconds while an optically-detected magnetic resonance
(ODMR) experiment is running. The reported temperature for each data point is
the average of the readings over the last 10 minutes of the experiment. The
standard deviations of the temperatures recorded during this interval were
within 120 mK for all experiments. Peak-to-peak variations were within 500 mK.
Because the temperature stability becomes worse as the temperature increases
from the base temperature in either operation mode, these values indicate the
stability at the highest temperatures reached in either low- or high-
temperature operation mode. In both modes, temperatures were roughly $4\times$
more stable at the lower end of the mode’s temperature range versus the higher
end. In order to minimize the temperature difference between the sample and
the sensor, microwaves were delivered by a wire loop (AWG 36 magnet wire, loop
radius ${\sim}1.5$ mm) suspended above the diamond sample.

The pulsed ODMR sequence performed in this work consists of optical
polarization (${\sim}3$ mW at 532 nm for 2 $\mathrm{\SIUnitSymbolMicro s}$)
and a microwave $\pi$ pulse (variable frequency, $200$ ns) before optical spin
state readout (${\sim}3$ mW at 532 nm for ${\sim}400$ $\mathrm{ns}$). The
sequence was repeated several hundred thousand times to sufficiently average
down the signal. The coordinates of the NV being measured were updated roughly
once every two minutes so that the NV remained in focus. The total measurement
time varied depending on how many repetitions of the sequence are required,
but was between 15 and 25 minutes.

The diamond sample used in this work was grown via chemical-vapor-deposition
by Element Six and displays a native, as-grown NV concentration of
approximately $10^{-3}$ ppb in each of the two regions that were studied. This
sample was previously used to characterize spin-lattice relaxation of the NV
center ground-state electronic spin in Ref. [1], where it was identified as
“Sample B.”

A potential systematic in temperature dependence studies is measurement-
induced heating [2, 3], in which the measurement itself places an additional
heat load on the sample. If this load is not shared by the temperature sensor,
then the resulting temperature difference between the sample and the sensor is
a systematic error. We characterized measurement-induced heating by running
the ODMR pulse sequence with the laser and microwaves focused directly on the
temperature sensor next to the diamond sample in low-temperature operation
mode where the heating effect is most significant. At 100 K, the sensor
registered a temperature increase of ${\sim}0.6$ K. At 200 K, this was reduced
to ${\sim}0.3$ K, and at 286 K the heating was ${\sim}0.1$ K. The rise time of
the heating once measurement began was ${\sim}10$ seconds, much shorter than
the total measurement duration of at least 15 minutes. While this heating
effect is partially accounted for by having the temperature sensor very close
to the diamond sample, the effect likely induces a small temperature
difference between the sensor and the diamond. We can bound the magnitude of
this systematic by assuming that only the diamond sample experiences
measurement-induced heating and that the reading of the temperature sensor is
the same whether or not a measurement is underway on the adjacent diamond
sample. Using the fit model of the ZFS temperature dependence from the main
text, we calculate that a +0.6 K temperature change at 100 K induces a -2 kHz
ZFS shift, a +0.3 K temperature change at 200 K induces a -11 kHz ZFS shift
and a +0.1 K temperature change at 286 K induces a -7 kHz ZFS shift. These
values are all nearly an order of magnitude below the typical statistical
errors recorded for a given measurement. For completeness we note that heating
effects below 100 K induce a negligible ZFS shift according to our model. For
example, a +10 K temperature change at 50 K induces a ZFS shift below 1 kHz.

Figure S1:  Near-room temperature zero-field ODMR survey of NV centers from
six different regions separated by hundreds of microns in the same diamond
sample. All NVs are several tens of microns below the diamond surface. NV
centers from region 1 (2) are the set A (B) NVs introduced in the main text.
Centers from regions 3-6 were not measured over a broad temperature range. NVs
in regions 2-6 were measured under ambient conditions (296 K) in high-
temperature operation mode. NVs in region 1 were measured at 295.1 K in low-
temperature operation mode. To account for this difference in temperature, the
deviations for region 1 NVs are shifted by -62 kHz, which is the difference in
ZFS predicted by the model described in the main text between 295.1 and 296 K.
After completion of measurements in the cryostat, we attempted to relocate the
set A NV centers so that we could re-characterize them under ambient
conditions, but were ultimately unable to find them. Note that splittings
below the ODMR linewidth (${\sim}5$ MHz) are possibly artifacts of the ODMR
spectrum fit. Visual inspection of the ODMR data sets with splittings below 5
MHz do not display two clearly resolved peaks. At larger splittings, however,
two distinct peaks are apparent.

##  II ODMR spectrum fitting

Each ODMR spectra is fit with a sum of two identical Voigt profiles that have
different center frequencies, which we refer to as a double-Voigt profile. A
Voigt profile is the convolution of a Gaussian and a Lorentzian. The
parameters that define a single Voigt profile are contrast, center frequency,
Gaussian width, and Lorentzian width. If the Gaussian (Lorentzian) width is
set to zero, then the Voigt profile is equivalent to a Lorentzian (Gaussian)
profile. In principle, because pulsed ODMR is coherent, the ODMR spectrum
could be fit by solving for the NV center’s Rabi lineshape under zero magnetic
field. In practice, this is complicated by the several physical factors that
affect the lineshape, including the relevance of all three NV levels at zero
field, hyperfine interactions, local electric field and strain, and microwave
polarization. We find that the fully empirical and comparatively simple
double-Voigt profile provides an excellent description of all the data
collected for this work. The profile is parameterized in terms of the ZFS $D$
and a splitting $S$, such that the center frequency of each single Voigt
profile is given by $D\pm S/2$.

##  III Variation of zero-field ODMR spectra from NV centers in different
regions of the diamond sample

As described in the main text, the ZFS of the NV centers in set A is on
average roughly 300 kHz lower than the ZFS of set B NVs near room temperature.
To investigate whether this may be due to regional variations in strain or the
local density of trapped charges, we recorded zero-field ODMR on NV centers
from five different regions in the diamond under ambient conditions. In
addition to characterizing the shifts in the ZFS, we also looked for
splittings of the ODMR line that would similarly indicate local strain or
electric fields that could lead to ZFS shifts. The results are displayed in
Fig. S1. We see that NV centers in set A (region 1 in Fig. S1) display a
relatively large spread in their zero-field splittings near room temperature
relative to centers from other regions. We note that this does not necessarily
reflect a real shift in the ZFS. An apparent shift could be caused by the
presence of a dark state induced by a combination of local electric/strain
fields and microwave polarization [4]. Besides the variation in the set A NV
centers, we also observe that NV centers in region 5 had consistently larger
splittings in their zero-field ODMR lines relative to NV centers in other
regions. These observations suggest that there are likely locations within the
diamond with higher concentrations of charged defects and/or local strain.
This is also consistent with the large variations in the density of native NV-
centers throughout the diamond sample.

##  IV Reproduction of prior results and additional comparisons

In Fig. 4 of the main text we reproduce the models from prior works that have
examined the ZFS temperature dependence over at least a 100 K temperature
range [5, 6, 7, 8, 9] and compare them to the model presented in this work. An
additional work, Ref. [10], measured the ZFS between 75 and 400 K, but did not
publish their fit parameters and so was not included in the main text figure.
In this section we review these prior models and experimental data sets in
more detail. We note that two of the prior works do not consider the absolute
value of the ZFS, and instead only present the deviation of the ZFS from its
zero-temperature limit [7, 9]. In order to compare the data and models from
these works to other absolute ZFS results, we add a constant offset equal to
the zero-temperature limit ZFS found in this work, $D_{0}=2877.38(3)$ MHz.

###  IV.1 Reproduction of prior models

The fit from each prior work is reproduced according to the model function and
fit parameters published in that work. The fit functions and original fit
parameters are presented in Table S1. The fit functions from Refs. [5] and [6]
are purely empirical fifth- and third-order polynomials respectively. Doherty
et al. motivate some parts of their model with arguments similar to those
invoked in this work and in Ref. [11], observing that the ZFS can be modeled
by summing contributions from thermal expansion and electron-phonon
interactions [7]. They calculate the thermal expansion contribution directly
using prior results describing the diamond coefficient of thermal expansion
[12] and the ZFS response to hydrostatic pressure [13]. Doherty et al. argue
that electron-phonon interactions contribute only starting at fourth order in
temperature such that the final ZFS temperature dependence expression from
Doherty et al. is a fifth-order polynomial with two free parameters that
describe the electron-phonon contributions. The expression does not include a
constant term or linear terms and only models the deviation of the ZFS from
the zero-temperature limit. Li et al. present a fit based on a modified
Varshni equation [8]. The equation transitions from quartic behavior at low
temperatures to quadratic behavior at higher temperatures and so is similar to
a polynomial in practice. The model presented by Barson et al. [9] builds on
that developed in Ref. [7]. Instead of using a power series to model the
coefficient of thermal expansion, Barson et al. use the expression from Ref.
[14], which consists of a weighted sum of Einstein heat capacities as in Eq. 8
of the main text of this work. Barson et al. additionally add a sixth-order
term to the polynomial, resulting in a final expression with three free
parameters. They fit their model to an artificial data set covering 0-700 K,
which is generated from prior works [7, 6]. The data sets from these two prior
works feature a discontinuity at room temperature, which is manually removed,
and the final artificial data set only describes the deviation of the ZFS from
the zero-temperature limit. Finally, Lourette et al. state that they fit their
data using a fourth-order polynomial, but do not provide the extracted fit
parameters in their manuscript, and so we do not attempt to reproduce their
fit here [10].

###  IV.2 Comparison to prior experimental data

Figure S2:  Comparison to prior experimental measurements of the ZFS
temperature dependence. Prior experimental data is digitized and reproduced
from Refs. [6, 5, 7, 8, 10]. Inset: Low-temperature behavior of data.

Fig. S2 displays the experimental data collected for this work alongside data
from prior works [5, 6, 7, 8, 10]. The experimental data was digitized from
Fig. 3a in Ref. [5], Fig. 5c in Ref. [6], Fig. 2a in Ref. [7], Fig. 1b in Ref.
[8], and Fig. 3e (black data points for 14NV) in Ref. [10]. The data was
digitized using WebPlotDigitizer [15]. Data from Refs. [5, 7, 8, 10] is for NV
ensembles while data from Ref. [6] is for a single NV center. We note that Li
et al. included additional data sets for three single NVs each from the same
sample which we do not reproduce in this work [8]. These single NV data sets
are relatively noisy and exhibit ZFS values that are unusually high, lying
roughly 1 MHz above the values reported by Li et al. for their ensemble
measurements. Li et al. speculate that this may be due to impurities in the
diamond sample used for single NV measurements [8]. Lourette et al. report
additional ZFS measurements of NV ensembles in 15N-enriched diamond samples
[10]. We do not reproduce these results here, but note that Lourette et al.
observe a small ZFS isotope shift of about +120 kHz in 15NV centers.

The available prior data sets of the NV center ZFS temperature dependence
disagree in several ways, as seen in Fig S2. Below room temperature, the data
and fits from Refs. [5] and [8] exhibit very similar ZFS values, which are
approximately 300 kHz above the ZFS values measured in this work. It is likely
that this difference is a sample dependent effect, as the temperature
dependence of the ZFS is negligible between 0 and 50 K and the difference
between our measurements and the measurements from Refs [5] and [8] is
consistent up to room temperature. The variation of the ZFS described in Sec.
III provides additional evidence that this is a plausible explanation. Near
room temperature, we observe that the ZFS of set B NVs (which were used for
high temperature measurements) is similarly about 300 kHz above the ZFS of set
A NVs (which were used for low temperature measurements). The set B NV
measurements are consistent with other prior measurements near room
temperature.

At higher temperatures, we also observe variations between the different works
in the measured slopes of the ZFS temperature dependence. While near room
temperature our measurements (for set B NVs) and the measurements from Refs.
[6] and [10] agree to within 100 kHz, by 400 K, all three measurements differ
significantly. Lourette et al.’s measurement is approximately 500 kHz below
the measurement made for this work [10], and Toyli et al.’s measurement is
approximately 750 kHz below Lourette et al.’s measurement. Toyli et al.’s
measurement is approximately 1.25 MHz below our measurement [6] at the same
temperature. We suggest that these differences are likely due to systematic
errors in measuring the temperature of the diamond at high temperatures.

Figure S3:  Pairwise comparisons between this work and a given prior work. In
each panel, the model and data set from a prior work is plotted alongside the
model from this work (solid and dashed blue line). Note that the model
presented by Barson et al. in Ref. [9] is fit to artificial data generated
from Refs. [7, 6]. As such, there is no experimental data shown in panel (e).
Lourette et al. do not provide the fit parameters for the fourth-order
polynomial they use to fit to their data set so only the data from Ref. [10]
is shown in panel (f). For panels (c) and (e), in accordance with what was
published in Refs. [7] and [9], the ZFS deviation from the zero-temperature
limit is shown rather than the absolute ZFS. As in the main text, the solid
fit lines become dashed outside the range of experimental data used for the
fit. For clarity, the experimental data from this work is not replotted.

###  IV.3 Pairwise comparisons

Fig. 4 in the main text and Fig. S2 compare the prior models and data sets
respectively from this prior works on the same plot. Fig. S3 presents the same
data in a pairwise format so as to allow for easier comparison between this
work and a specific prior work.

###  IV.4 Fits of prior models to experimental data collected for this work

In this section we compare the quality of the fits provided by the model
proposed in this work and the models from prior works, by fitting the models
to the same data set. Because goodness-of-fit metrics are strongly affected by
the 300 kHz offset between the set A and B measurements, we fit the models to
ZFS deviation data generated by subtracting the zero-temperature ZFS limit
given in the main text from each of the set A and B measurements performed for
this work, and an additional 300 kHz from the set B measurements to get rid of
the offset. The resulting fits are presented in Table S1 and Fig. S4. Our
model and the fifth order power series used by Chen et al. [5] both provide
excellent fits to the data, with reduced chi squared metrics $\chi_{\nu}^{2}$
of 1.50. While the power series from Chen et al. provides a good fit over the
temperature range probed in this work, we emphasize that it is not suitable
for predicting the ZFS at higher temperatures, and that it would perform worse
if used to fit ZFS data covering a broader temperature range, as suggested by
the high temperature ZFS measurements made by Toyli et al. [6]. The model from
Barson et al. [9] is slightly worse according to its reduced chi squared of
$\chi_{\nu}^{2}=2.23$. Our model and the models from Refs. [5, 9] provide fits
to the data that are visually similar. The models from Refs. [6, 7, 8] provide
relatively poor fits with high reduced chi squared metrics and visually
apparent departures from the experimental data.

###  IV.5 Fits of model proposed in this work to experimental data from prior
works

Because of the discrepancies between the ZFS data collected for this work and
for prior works, the model and fit parameters described in the main text is
not able to describe all the available data simultaneously. Here we
demonstrate that a version of the model with fixed representative phonon mode
energies can provide an excellent description of all the available data by
allowing the coefficients and zero-temperature ZFS limit to be free
parameters. We fix the mode energies to the values reported in the main text,
$\Delta_{1}=58.73$ meV and $\Delta_{2}=145.5$ meV. Fig. S5 shows the resulting
fits to the original data sets from Refs. [6, 5, 7, 8, 10]. Despite the
discrepancies between the different data sets, we see that the model with
fixed energies does an excellent job of describing each individual data set.
This indicates that the success of the model is not specific to the data set
collected for this work, and likewise that the representative phonon mode
energies reported in the main text are not specific to any given NV or sample.

Reference |  Functional form |  Fit free
parameters |  Original fit |  Refit to ZFS deviation data from this work |  Refit $\chi_{\nu}^{2}$
---|---|---|---|---|---
This work |  $D(T)=D_{0}+c_{1}n_{1}+c_{2}n_{2}$ |  $D_{0}$ (MHz) |  2877.38(3) |  n/a |  1.50
|  |  $c_{1}$ (MHz) |  -55(7) |  $-75(6)$ |
|  |  $\Delta_{1}$ (meV) |  59(2) |  63.9(15) |
|  |  $c_{2}$ (MHz) |  $-2.5(19)\text{\times}{10}^{2}$ |  $-2.5(3)\text{\times}{10}^{2}$ |
|  |  $\Delta_{2}$ (meV) |  146(8) |  163(12) |
[5] Chen et al. |  $D(T)=\sum_{i=0}^{5}d_{i}T^{i}$ |  $d_{0}$ (MHz) |  2877.71 |  n/a |  1.50
|  |  $d_{1}$ (MHz/K) |  $-4.625\text{\times}{10}^{-3}$ |  $-5.6(14)\text{\times}{10}^{-3}$ |
|  |  $d_{2}$ (MHz/K2) |  $1.067\text{\times}{10}^{-4}$ |  $1.6(2)\text{\times}{10}^{-4}$ |
|  |  $d_{3}$ (MHz/K3) |  $-9.325\text{\times}{10}^{-7}$ |  $-1.31(11)\text{\times}{10}^{-6}$ |
|  |  $d_{4}$ (MHz/K4) |  $1.739\text{\times}{10}^{-9}$ |  $2.3(2)\text{\times}{10}^{-9}$ |
|  |  $d_{5}$ (MHz/K5) |  $-1.838\text{\times}{10}^{-12}$ |  $-1.48(18)\text{\times}{10}^{-12}$ |
[6] Toyli et al. |  $D(T)=\sum_{i=0}^{3}a_{i}T^{i}$ |  $a_{0}$ (MHz) |  2869.7(9) |  n/a |  9.90
|  |  $a_{1}$ (MHz/K) |  $9.7(6)\text{\times}{10}^{-2}$ |  $1.65(3)\text{\times}{10}^{-2}$ |
|  |  $a_{2}$ (MHz/K2) |  $-3.7(1)\text{\times}{10}^{-4}$ |  $-1.274(18)\text{\times}{10}^{-4}$ |
|  |  $a_{3}$ (MHz/K3) |  $1.7(1)\text{\times}{10}^{-7}$ |  $-3.7(3)\text{\times}{10}^{-8}$ |
[7] Doherty et al. |  $\Delta D(T)=-AB\sum_{i=1}^{5}(e_{i-1}/i)T^{i}$ |  $b_{4}$ (MHz/K4) |  $18.7(4)\text{\times}{10}^{-10}$ |  $1.161(3)\text{\times}{10}^{-9}$ |  61.07
|  $+\sum_{i=4}^{5}b_{i}T^{i}$ |  $b_{5}$ (MHz/K5) |  -$41(2)\text{\times}{10}^{-13}$ |  $-1.932(6)\text{\times}{10}^{-12}$ |
[8] Li et al. |  $D(T)=D_{0}-[AT^{4}/(T+B)^{2}]$ |  $D_{0}$ (MHz) |  2877.69(1) |  n/a |  19.86
|  |  $A$ (MHz/K2) |  $5.6(4)\text{\times}{10}^{3}$ |  $2.029(10)\text{\times}{10}^{-4}$ |
|  |  $B$ (K) |  490(32) |  $168.8(15)$ |
[9] Barson et al. |  $\Delta D(T)=-AB(\Delta V/V)+\sum_{i=4}^{6}b_{i}T^{i}$ |  $b_{4}$ (MHz/K4) |  $-1.44(8)\text{\times}{10}^{-9}$ |  $-1.813(18)\text{\times}{10}^{-9}$ |  2.23
|  |  $b_{5}$ (MHz/K5) |  $3.1(3)\text{\times}{10}^{-12}$ |  $4.90(9)\text{\times}{10}^{-12}$ |
|  |  $b_{6}$ (MHz/K6) |  $-1.8(3)\text{\times}{10}^{-15}$ |  $-3.79(10)\text{\times}{10}^{-15}$ |
Table S1: Comparison of the functional forms and fit parameters of the
available models of the ZFS temperature dependence. Note that some works
originally report parameters in units of GHZ rather than MHz. For consistency,
we have converted to MHz where this is the case. Here we match the notation of
the functional forms used in the original works. The first terms in the
expressions from Refs. [7] and [9] describe the effect of thermal expansion
and are not free parameters of the fit. In these works the effect of thermal
expansion is expressed in terms of the NV hydrostatic pressure shift $A$, the
diamond bulk modulus $B$, and the diamond coefficient of thermal expansion.
Doherty et al. use values of $A=14.6$ MHz/GPa [13] and $B=442$ GPa for the
hydrostatic pressure shift and bulk modulus respectively. The coefficient of
thermal expansion terms $e_{i}$ are derived from Ref. [12] and are given by
Doherty et al. as $ABe_{1}/2=$39.7\text{\times}{10}^{-7}$$ MHz / T2,
$ABe_{2}/3=$-91.6\text{\times}{10}^{-9}$$ MHz / T3,
$ABe_{3}/4=$70.6\text{\times}{10}^{-11}$$ MHz / T4,
$ABe_{4}/5=$-60.0\text{\times}{10}^{-14}$$ MHz / T5. Barson et al. use the
same values as Doherty et al. for the hydrostatic pressure shift $A$ and the
bulk modulus $B$, and describe the coefficient of thermal expansion $\Delta
V/V$ using the expression from Ref. [14], which is a weighted sum over
Einstein heat capacities and is not given by Barson et al. in their
manuscript. In contrast, in Ref. [8] $A$ and $B$ refer to free parameters in
the fit. The last two columns in the table describe the fits of the models to
the ZFS deviation data collected for this work, as shown in Fig. S4. Figure
S4:  Fits of prior models to ZFS deviation data collected for this work. The
models from Refs. [6, 13, 8] clearly stray from the experimental data. The
other models provide reasonable fits to the data. The fit parameters and
reduced chi squared metrics are shown in Table S1.  Figure S5:  Fits of the
model proposed in this work to prior data sets. The energies of the
representative phonon modes in the model are fixed to the energies reported in
the main text. For fitting to the data from Ref. [7] there are two free
parameters, the coefficients $c_{1,2}$ in Eq. 3 of the main text. For fitting
to the data from the other references there is an additional free parameter,
$D_{0}$ the zero-temperature ZFS limit. Panel (a) is the same as Fig. 2 of the
main text and is reproduced here to allow for easier comparison. Because error
bars are not available for the prior data sets, we demonstrate the fit
qualities by fixing the range of the residuals shown in the bottom plot of
each panel to [-0.45, +0.45] MHz. The residuals across all temperatures and
data sets fall within this range, indicating that the model can provide an
excellent description of all prior data sets in addition to the data set
collected for this work. Note that the accuracy of the prior data sets may be
limited by the digitization procedure.

##  V Comparison of phonon energies identified in this work and prior works

In this work we describe the temperature dependence of the NV center ZFS by
considering the effects of two representative phonon modes. Several prior
works have used a similar technique to describe other properties of the NV
center that are physically rooted in interactions with phonons. In this
section we compare the various energies that are reported by these works. In
this work, the fit described in the main text yields energies of
$\Delta_{1}=59(2)$ meV and $\Delta_{2}=146(8)$. Fitting the same model to data
of the deviation of the ZFS from its zero-temperature limit (see Section IV.1,
Fig. S4, and Table S1 for details) we find slightly higher but consistent
energies of 63.9(15) and 163(12) meV. In Ref. [1] we used a similar model to
describe the temperature dependence of the NV spin-phonon relaxation rates.
The energies reported in that work are 68.2(17) and 167(12) meV. Studies of
the NV optical spectrum have noted that several prominent features of the
phonon sideband are associated with particular phonon energies. Davies’ model
of the sideband is based on a continuous electron-phonon coupling spectrum
with peaks at 69 and 154 meV [16]. Collins, Stanley, and Woods observe phonon
replicas of the zero-phonon line at spacings of 61 and 137 meV [17]. In
another experimental work, Kehaysias et al. identify significant features in
the phonon sideband at 64, 122, 138, 153, and 163 meV [18]. Alkauskas et al.
study the NV optical spectrum from first principles and identify a dominant 65
meV vibrational resonance, as well as additional modes at 120, 134, 161, and
167 meV that are highly localized to the NV center [19]. While some of the
energies identified in these works may be linked to individual phonon modes
that strongly influence NV center dynamics, other energies instead represent
the effect of several modes with similar but not identical energies. In this
latter case we interpret the identified energy as that of a single
“representative” phonon mode which stands in for the group of several modes.
We expect that such representative modes will tend to have a lower energy than
the typical energy of the group of modes it represents, as the lower energy
modes in the group have larger occupation numbers. This consideration, as well
as the fact that the various physical properties (e.g. the ZFS temperature
dependence, spin-phonon relaxation rate temperature dependence, the phonon
sideband) will be affected differently by the same spectrum of phonon modes,
indicates that the energies identified by examining one physical property
cannot be neatly compared to the energies identified by examining a different
physical property. Nevertheless, as we argue in the main text, the various
phonon-influenced properties of diamond and the NV center derive from the same
phonon density of states, and so it is reasonable to expect that different
studies will identify energies that roughly correspond to peaks (or are
somewhat lower than the peaks) in the phonon density of states. This claim is
generally true empirically, as can be seen by comparing the energies in the
survey reported here against the measured phonon density of states [20].

##  VI Generation of lattice constant data from Ref. [21]

In Ref. [21], Jacobson and Stoupin survey existing experimental studies of the
diamond coefficient of thermal expansion and lattice constant. Building off
prior work within the quasiharmonic model of thermal expansion, Jacobson and
Stoupin fit the various measurements of the coefficient of thermal expansion
to a weighted sum of Einstein heat capacities at four different frequencies.
They present two different fits using the same model where Fit 1 (Fit 2)
covers experimental data up to 1000 K (2000 K). While both fits provide an
excellent description of the experimental up to 1000 K, Fit 2 yields
activation energies above the energy of the debye temperature, inconsistent
with the quasiharmonic model. Jacobson and Stoupin further accurately
reproduce measurements of the lattice constant up to 1000 K by integrating
both fits of the coefficient of thermal expansion according to Eq. 9 from the
main text with $a_{0}=3.56712$ $\mathrm{\SIUnitSymbolAngstrom}$ at $T_{0}=298$
K [22]. NV center measurements have been demonstrated up to a maximum
temperature of 1000 K [23], and so for Fig. 3a of the main text we represent
the diamond lattice constant using the curve calculated from Fit 1 of Ref.
[21]. The fit parameters we find by fitting Eq. 11 of the main text to this
generated lattice constant data are
$a_{0}=3.56677$\mathrm{\SIUnitSymbolAngstrom}$$,
$b_{1}=$2.027\text{\times}{10}^{-03}$$\mathrm{\SIUnitSymbolAngstrom}$$ and
$b_{2}=$2.609\text{\times}{10}^{-02}$$\mathrm{\SIUnitSymbolAngstrom}$$.

##  VII Complete data set collected for this work

The complete set of experimental data collected for this work is presented in
Table S2.

NV |  $T$ (K) |  $D$ (MHz) | NV |  $T$ (K) |  $D$ (MHz) | NV |  $T$ (K) |  $D$ (MHz) | NV |  $T$ (K) |  $D$ (MHz) | NV |  $T$ (K) |  $D$ (MHz)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
A1 |  15.3 | 2877.36(12) | A2 |  15.3 | 2877.23(12) | A3 |  15.3 | 2877.52(10) | A4 |  15.3 | 2877.41(12) | A5 |  15.3 | 2877.21(14)
A1 |  26.0 | 2877.63(16) | A2 |  26.0 | 2877.44(15) | A3 |  26.0 | 2877.41(13) | A4 |  26.0 | 2877.27(16) | A5 |  26.0 | 2877.0(2)
A1 |  37.7 | 2877.5(3) | A2 |  37.7 | 2877.4(3) | A3 |  37.7 | 2877.0(3) | A4 |  37.7 | 2878.1(2) | A5 |  37.7 | 2877.9(4)
A1 |  45.8 | 2876.5(5) | A2 |  45.8 | 2877.9(4) | A3 |  45.8 | 2877.0(4) | A4 |  45.8 | 2877.6(4) | A5 |  45.8 | 2877.3(4)
A1 |  54.0 | 2877.1(4) | A2 |  54.0 | 2877.5(5) | A3 |  54.1 | 2877.6(3) | A4 |  54.1 | 2877.3(4) | A5 |  54.2 | 2877.4(3)
A1 |  62.9 | 2877.6(3) | A2 |  62.9 | 2877.4(3) | A3 |  63.0 | 2876.9(2) | A4 |  63.0 | 2877.4(2) | A5 |  63.0 | 2877.4(3)
A1 |  74.4 | 2877.5(3) | A2 |  74.4 | 2877.0(3) | A3 |  74.4 | 2877.3(2) | A4 |  74.4 | 2877.29(18) | A5 |  74.4 | 2877.2(2)
A1 |  85.2 | 2877.0(3) | A2 |  85.2 | 2877.8(3) | A3 |  85.2 | 2877.2(2) | A4 |  85.2 | 2877.31(18) | A5 |  85.2 | 2877.4(3)
A1 |  95.9 | 2877.35(18) | A2 |  95.9 | 2877.3(2) | A3 |  95.9 | 2877.27(20) | A4 |  95.8 | 2877.19(17) | A5 |  95.9 | 2877.20(18)
A1 |  106.5 | 2877.1(2) | A2 |  106.5 | 2877.4(2) | A3 |  106.5 | 2877.36(18) | A4 |  106.5 | 2877.17(17) | A5 |  106.5 | 2877.07(17)
A1 |  116.9 | 2877.6(2) | A2 |  116.9 | 2877.3(2) | A3 |  116.9 | 2877.03(16) | A4 |  116.9 | 2877.07(17) | A5 |  116.9 | 2877.24(18)
A1 |  127.1 | 2876.97(19) | A2 |  127.1 | 2877.3(2) | A3 |  127.1 | 2877.57(16) | A4 |  127.1 | 2876.95(15) | A5 |  127.0 | 2877.15(15)
A1 |  137.3 | 2877.44(17) | A2 |  137.2 | 2876.93(17) | A3 |  137.3 | 2877.11(15) | A4 |  137.3 | 2876.87(15) | A5 |  137.3 | 2877.18(17)
A1 |  147.3 | 2876.71(15) | A2 |  147.3 | 2876.99(16) | A3 |  147.3 | 2876.60(15) | A4 |  147.3 | 2876.80(14) | A5 |  147.3 | 2877.13(15)
A1 |  157.3 | 2876.61(18) | A2 |  157.3 | 2876.64(17) | A3 |  157.3 | 2876.47(14) | A4 |  157.4 | 2876.66(14) | A5 |  157.4 | 2876.55(14)
A1 |  167.3 | 2876.54(15) | A2 |  167.3 | 2876.51(18) | A3 |  167.3 | 2876.59(17) | A4 |  167.4 | 2876.15(13) | A5 |  167.3 | 2876.53(13)
A1 |  177.3 | 2876.27(15) | A2 |  177.3 | 2876.16(17) | A3 |  177.3 | 2876.01(15) | A4 |  177.3 | 2876.22(13) | A5 |  177.3 | 2875.97(13)
A1 |  187.2 | 2876.13(14) | A2 |  187.2 | 2876.07(17) | A3 |  187.2 | 2875.96(13) | A4 |  187.2 | 2875.56(12) | A5 |  187.2 | 2875.83(14)
A1 |  197.1 | 2875.49(18) | A2 |  197.1 | 2876.01(17) | A3 |  197.1 | 2875.53(15) | A4 |  197.1 | 2875.38(12) | A5 |  197.1 | 2875.37(13)
A1 |  207.2 | 2875.12(17) | A2 |  207.1 | 2875.49(16) | A3 |  207.2 | 2875.48(14) | A4 |  207.2 | 2875.21(13) | A5 |  207.1 | 2875.06(13)
A1 |  217.2 | 2874.90(16) | A2 |  217.2 | 2875.14(18) | A3 |  217.2 | 2874.85(14) | A4 |  217.2 | 2874.67(13) | A5 |  217.2 | 2874.35(16)
A1 |  227.1 | 2874.64(19) | A2 |  227.1 | 2874.56(17) | A3 |  227.1 | 2874.34(15) | A4 |  227.1 | 2874.26(13) | A5 |  227.1 | 2874.07(15)
A1 |  237.6 | 2873.72(19) | A2 |  237.6 | 2874.0(2) | A3 |  237.6 | 2873.89(14) | A4 |  237.6 | 2873.69(12) | A5 |  237.6 | 2873.65(15)
A1 |  247.3 | 2873.25(18) | A2 |  247.2 | 2873.52(19) | A3 |  247.2 | 2873.22(14) | A4 |  247.2 | 2873.29(14) | A5 |  247.2 | 2873.21(13)
A1 |  257.1 | 2872.81(20) | A2 |  257.0 | 2873.11(18) | A3 |  257.1 | 2872.98(14) | A4 |  257.1 | 2872.75(13) | A5 |  257.0 | 2872.87(17)
A1 |  267.3 | 2872.36(15) | A2 |  267.2 | 2872.02(18) | A3 |  267.2 | 2871.90(14) | A4 |  267.2 | 2871.87(15) | A5 |  267.3 | 2872.18(14)
A1 |  276.4 | 2871.77(15) | A2 |  276.4 | 2871.80(18) | A3 |  276.4 | 2871.44(15) | A4 |  276.4 | 2871.45(14) | A5 |  276.4 | 2871.50(14)
A1 |  285.9 | 2870.77(14) | A2 |  285.9 | 2870.98(17) | A3 |  285.9 | 2870.72(15) | A4 |  286.0 | 2870.74(14) | A5 |  286.0 | 2870.74(14)
A1 |  295.1 | 2870.77(19) | A2 |  295.2 | 2870.5(2) | A3 |  295.2 | 2870.40(14) | A4 |  295.1 | 2870.20(13) | A5 |  295.1 | 2870.17(15)
B1 |  296.0 | 2870.53(8) | B2 |  296.0 | 2870.67(8) | B3 |  296.0 | 2870.74(7) | B4 |  296.0 | 2870.57(9) | B5 |  296.0 | 2870.72(8)
B1 |  310.0 | 2869.39(8) | B2 |  310.0 | 2869.48(8) | B3 |  310.0 | 2869.58(7) | B4 |  310.0 | 2869.64(11) | B5 |  310.0 | 2869.64(10)
B1 |  320.0 | 2868.75(9) | B2 |  320.0 | 2868.66(9) | B3 |  320.0 | 2868.80(8) | B4 |  320.0 | 2868.65(11) | B5 |  320.0 | 2868.84(9)
B1 |  330.0 | 2867.96(7) | B2 |  330.0 | 2867.67(8) | B3 |  330.0 | 2867.93(7) | B4 |  330.0 | 2868.09(9) | B5 |  330.0 | 2868.16(8)
B1 |  340.0 | 2867.18(7) | B2 |  340.0 | 2867.06(8) | B3 |  340.0 | 2867.07(8) | B4 |  340.0 | 2866.98(9) | B5 |  340.0 | 2867.03(8)
B1 |  350.0 | 2866.24(8) | B2 |  350.0 | 2866.15(9) | B3 |  350.0 | 2866.38(9) | B4 |  350.0 | 2866.28(11) | B5 |  350.0 | 2866.41(10)
B1 |  360.0 | 2865.24(8) | B2 |  360.0 | 2865.26(9) | B3 |  360.0 | 2865.37(8) | B4 |  360.0 | 2865.50(11) | B5 |  360.0 | 2865.44(10)
B1 |  370.0 | 2864.61(8) | B2 |  370.0 | 2864.43(8) | B3 |  370.0 | 2864.51(8) | B4 |  370.0 | 2864.58(11) | B5 |  370.0 | 2864.31(11)
B1 |  380.0 | 2863.56(9) | B2 |  380.0 | 2863.23(9) | B3 |  380.0 | 2863.14(8) | B4 |  380.0 | 2863.56(14) | B5 |  380.0 | 2863.53(11)
B1 |  390.0 | 2862.40(9) | B2 |  390.0 | 2862.31(10) | B3 |  390.0 | 2862.57(8) | B4 |  390.0 | 2862.24(12) | B5 |  390.0 | 2862.48(12)
B1 |  400.0 | 2861.63(9) | B2 |  400.0 | 2861.50(10) | B3 |  400.0 | 2861.31(9) | B4 |  400.0 | 2861.37(15) | B5 |  400.0 | 2861.60(14)
B1 |  410.0 | 2860.17(9) | B2 |  410.0 | 2860.34(11) | B3 |  410.0 | 2860.34(10) | B4 |  410.0 | 2860.12(9) | B5 |  410.0 | 2860.38(9)
B1 |  420.0 | 2859.46(10) | B2 |  420.0 | 2859.10(11) | B3 |  420.0 | 2859.42(10) | B4 |  420.0 | 2859.41(11) | B5 |  420.0 | 2859.33(9)
B1 |  430.0 | 2858.30(9) | B2 |  430.0 | 2858.17(11) | B3 |  430.0 | 2858.27(9) | B4 |  430.0 | 2858.40(12) | B5 |  430.0 | 2858.32(9)
B1 |  440.0 | 2856.93(9) | B2 |  440.0 | 2856.89(9) | B3 |  440.0 | 2857.25(10) | B4 |  440.0 | 2857.05(11) | B5 |  440.0 | 2856.96(9)
B1 |  450.0 | 2855.78(10) | B2 |  450.0 | 2855.78(11) | B3 |  450.0 | 2855.93(10) | B4 |  450.0 | 2856.22(12) | B5 |  450.0 | 2856.02(11)
B1 |  460.0 | 2855.03(11) | B2 |  460.0 | 2854.72(12) | B3 |  460.0 | 2854.89(11) | B4 |  460.0 | 2854.76(14) | B5 |  460.0 | 2854.59(12)
B1 |  470.0 | 2853.75(11) | B2 |  470.0 | 2853.86(12) | B3 |  470.0 | 2853.41(10) | B4 |  470.0 | 2853.64(17) | B5 |  470.0 | 2853.55(12)
B1 |  480.0 | 2852.14(12) | B2 |  480.0 | 2852.07(13) | B3 |  480.0 | 2852.19(10) | B4 |  480.0 | 2852.59(14) | B5 |  480.0 | 2852.23(12)
B1 |  490.0 | 2851.12(7) | B2 |  490.0 | 2850.94(9) | B3 |  490.0 | 2850.95(7) | B4 |  490.0 | 2850.82(13) | B5 |  490.0 | 2850.89(13)
B1 |  500.0 | 2849.93(9) | B2 |  500.0 | 2849.34(10) | B3 |  500.0 | 2849.67(8) | B4 |  500.0 | 2849.73(12) | B5 |  500.0 | 2849.62(14)
Table S2: Complete set of experimental measurements of zero-field splitting
$D$ versus temperature $T$ collected for this work. NVs are indexed first by
group and then numerically to distinguish single NVs.

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# Divide bounded sets into sets having smaller diameters

Yanlu Lian<EMAIL_ADDRESS>and  Senlin Wu<EMAIL_ADDRESS>
###### Abstract.

For each positive integer $m$ and each real finite dimensional Banach space
$X$, we set $\beta(X,m)$ to be the infimum of $\delta\in(0,1]$ such that each
set $A\subset X$ having diameter $1$ can be represented as the union of $m$
subsets of $A$ whose diameters are at most $\delta$. Elementary properties of
$\beta(X,m)$, including its stability with respect to $X$ in the sense of
Banach-Mazur metric, are presented. Two methods for estimating $\beta(X,m)$
are introduced. The first one estimates $\beta(X,m)$ using the knowledge of
$\beta(Y,m)$, where $Y$ is a Banach space sufficiently close to $X$. The
second estimation uses the information about $\beta_{X}(K,m)$, the infimum of
$\delta\in(0,1]$ such that $K\subset X$ is the union of $m$ subsets having
diameters not greater than $\delta$ times the diameter of $K$, for certain
classes of convex bodies $K$ in $X$. In particular, we show that
$\beta(l_{p}^{3},8)\leq 0.925$ holds for each $p\in[1,+\infty]$ by applying
the first method, and we proved that $\beta(X,8)<1$ whenever $X$ is a three-
dimensional Banach space satisfying $\beta_{X}(B_{X},8)<\frac{221}{328}$,
where $B_{X}$ is the unit ball of $X$, by applying the second method. These
results and methods are closely related to the extension of Borsuk’s problem
in finite dimensional Banach spaces and to C. Zong’s computer program for
Borsuk’s conjecture.

##  1\. Introduction

Let $X=(\mathbb{R}^{n},\left\|\cdot\right\|)$ be an $n$-dimensional Banach
space with unit ball $B_{X}$. For each $A\subset X$, we denote by
$\operatorname{int}{A}$ and $\operatorname{bd}{A}$ the _interior_ and the
_boundary_ of $A$, respectively. When $A$ is nonempty and bounded, we denote
by $\delta\left(A\right)$ the _diameter_ of $A$. I.e.,
$\delta\left(A\right)=\sup\quantity{\left\|x-y\right\|\mid x,y\in A}$. A
compact convex subset of $X$ having interior points is called a _convex body_.
Since any two norms on $\mathbb{R}^{n}$ induce the same topology, $K$ is a
convex body (is bounded, resp.) in $X$ if and only if it is a convex body (is
bounded, resp.) in $\mathbb{E}^{n}=(\mathbb{R}^{n},\left\|\cdot\right\|_{E})$,
where $\left\|\cdot\right\|_{E}$ is the regular Euclidean norm. Let
$\mathcal{K}^{n}$ be the set of all convex bodies in $\mathbb{E}^{n}$. A
bounded subset $A$ of $X$ is said to be _complete_ if $x\not\in
A\Rightarrow\delta\left((A\cup\quantity{x})\right)>\delta\left(A\right)$. It
is clear that each complete set is closed and convex. For each bounded subset
$A$ of $X$, there always exists a complete set $A^{c}$, called a _completion_
of $A$, with diameter $\delta\left(A\right)$ containing $A$. Note that $A$ may
have different completions. Put

$\mathcal{B}^{n}=\quantity{A\subset\mathbb{R}^{n}\mid\text{$A$ is bounded and
}\delta\left(A\right)>0},\quad\mathcal{C}_{X}=\quantity{A\in\mathcal{B}^{n}\mid
A\text{ is complete}}.$

In 1933, Borsuk [5] proposed the following:

######  Problem 1 (Borsuk’s Problem).

Is it possible to divide every bounded subset of $\mathbb{E}^{n}$ into $n+1$
sets having smaller diameters?

The answer is affirmative when $n\leq 3$ (cf. [10], [14], [17], or [16]), is
negative in each of the following cases: $n\geq 298$ (cf. [18]), $n\geq 65$
(cf. [4]), and $n=64$ (cf. [20]), and is not clear in other cases.

Progesses have been made by providing upper bounds for
$b(n)=\sup\quantity{b(A)\mid A\in\mathcal{B}^{n}}$, where $b(A)$, called the
_Borsuk number of $A$_, is the minimal positive integer $m$ such that $A$ is
the union of $m$ subsets having smaller diameter. For example, L. Danzer (cf.
[9]), M. Lassak (cf. [22]), and O. Schramm (cf. [26] and [6]) showed that

$b(n)\leq\sqrt{\frac{(n+2)^{3}(2+\sqrt{2})^{n-1}}{3}},\quad b(n)\leq
2^{n-1}+1,\mbox{\quad and\quad}b(n)\leq 5n^{\frac{3}{2}}(4+\log
n)\quantity(\frac{3}{2})^{\frac{n}{2}},$

respectively.

One can also study Problem 1 via estimating
$\beta(n,m)=\sup\quantity{\beta(A,m)\mid A\in\mathcal{B}^{n}}$, where $m$ is a
positive integer and, for each $A\in\mathcal{B}^{n}$,

$\beta(A,m)=\inf\quantity{\frac{1}{\delta\left(A\right)}\max\limits_{k\in[m]}\delta\left(A_{k}\right)\mid
A=\bigcup\limits_{k=1}^{m}A_{k}}.$

Here we used the shorthand notation $[m]:=\quantity{i\in\mathbb{Z}^{+}\mid
1\leq i\leq m}$.

In 2020, C. Zong (cf. [32]) proved that $\beta(A,m)$ is uniformly continuous
on the space $\mathcal{K}^{n}$ endowed with the Hausdorff metric, and
reformulated Problem 1 as the following:

######  Problem 2.

Does there exists a positive number $\alpha_{n}<1$ such that

$\beta(K,n+1)\leq\alpha_{n},~{}\forall K\in\mathcal{D}^{n},$

where

$\mathcal{D}^{n}=\quantity{K\in\mathcal{K}^{n}\mid B_{2}^{n}\subseteq
K\subseteq\frac{\sqrt{n/(2n+2)}}{1-\sqrt{n/(2n+2)}}B_{2}^{n}}?$

Some known estimations of $\beta(n,m)$ are listed in Table 1 below (cf., [14],
[21], [12], and [13]).

$n$ $m$ $\beta(n,m)$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$
---|---|---|---|---|---|---
$2$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{2}}{2}$ | $\geq\sin\frac{\pi}{5}$ | $\leq\frac{\sqrt{3}}{3}$ | $\frac{1}{2}$ | ?
$3$ | $-$ | $\in[0.8880,0.9887]$ | $\in\quantity[\frac{\sqrt{3}}{2},0.9425]$ | $\geq\frac{\sqrt{6}}{3}$ | $\geq\frac{\sqrt{2}}{2}$ | $\geq\frac{\sqrt{2}}{2}$
Table 1. Known estimations on $\beta(n,m)$

Grünbaum extended Borsuk’s problem into Banach spaces and asked the following
(cf. [15]):

######  Problem 3.

Let $A\subset X=(\mathbb{R}^{n},\left\|\cdot\right\|)$. What is the smallest
positive integer $m$, denoted by $b_{X}(A)$, such that $A$ can be represented
as the union of $m$ sets having smaller diameter.

Put

$\displaystyle b(X)=\max\quantity{b_{X}(A)\mid
A\in\mathcal{B}^{n}}=\max\quantity{b_{X}(A)\mid A\in\mathcal{K}^{n}},$
$\displaystyle
B(n)=\max\quantity{b((\mathbb{R}^{n},\left\|\cdot\right\|))\mid\left\|\cdot\right\|\text{
is a norm on $\mathbb{R}^{n}$}}.$

It is clear that

(1) $b_{X}(K)\leq c(K),~{}\forall K\in\mathcal{K}^{n},$

where $c(K)$ is the least number of smaller homothetic copies of $K$ needed to
cover $K$. Since Hadwiger’s covering conjecture (see, e.g., [3], [24], [8],
[31], [1]) asserts that $c(K)\leq 2^{n},~{}\forall K\in\mathcal{K}^{n}$, it is
reasonable to make the following conjecture (cf. [2, p. 75]):

######  Conjecture 1.

For each integer $n\geq 3$, $B(n)=2^{n}$.

When $X$ is two-dimensional and $A\in\mathcal{B}^{2}$,
$b_{X}(A)\in\quantity{2,3,4}$ (cf. [3, §33]). Therefore, $B(2)=4$. L. Yu and
C. Zong [30] proved that

(2) $b(l_{p}^{3})\leq 8,~{}\forall p\in[1,+\infty].$

By the main result of [19] and (1), there exist universal constants $c_{1}$
and $c_{2}>0$ such that $B(n)\leq c_{1}4^{n}e^{-c_{2}\sqrt{n}},~{}\forall
n\geq 2$. Despite of these progress, Conjecture 1 is still open when $n\geq
3$.

In this paper, we study Conjecture 1 by estimating

$\beta_{X}(A,m)=\inf\quantity{\frac{1}{\delta\left(A\right)}\max\quantity{\delta\left(A_{k}\right)\mid
k\in[m]}\mid A=\bigcup\limits_{k=1}^{m}A_{k}}$

for $A\in\mathcal{B}^{n}$, and

$\beta(X,m)=\sup\quantity{\beta_{X}(A,m)\mid A\in\mathcal{B}^{n}}.$

We focus mainly, but not only, on the case $n=3$.

In Section 2, we present elementary properties of $\beta(X,m)$, including its
stability with respect to $X$ in the sense of Banach-Mazur metric. In Section
3, we provide an estimation of $\beta(X,8)$ for three-dimensional Banach
spaces $X$ such that $\beta_{X}(B_{X},8)$ is sufficiently small. In Section 4,
we show that

$\beta(l_{p}^{3},8)\leq 0.925,~{}\forall p\in[1,+\infty],$

which can be viewed as a quantitative version of Yu and Zong’s result (2).

##  2\. Elementary properties of $\beta(X,m)$

######  Proposition 1.

For each finite dimensional Banach space
$X=(\mathbb{R}^{n},\left\|\cdot\right\|)$ and each positive integer $m$, we
have

$\beta(X,m)=\sup\quantity{\beta_{X}(A,m)\mid A\in\mathcal{C}_{X}}.$

###### Proof.

Put $\beta=\sup\quantity{\beta_{X}(A,m)\mid A\in\mathcal{C}_{X}}$. We only
need to show that $\beta(X,m)\leq\beta$. Let $A$ be an arbitrary set in
$\mathcal{B}^{n}$, and $A^{c}$ be a completion of $A$. For each
$\varepsilon>0$, there exists a collection $\quantity{B_{i}\mid i\in[m]}$ of
subsets of $A^{c}$ such that

$A^{c}=\bigcup\limits_{i\in[m]}B_{i},\mbox{\quad
and\quad}\frac{1}{\delta\left(A\right)}\max\quantity{\delta_{X}(B_{k})\mid
k\in[m]}\leq\beta_{X}(A^{c},m)+\varepsilon.$

It follows that

$A=A\cap A^{c}=\bigcup\limits_{i\in[m]}\quantity(A\cap B_{i}).$

Thus

$\beta_{X}(A,m)\leq\frac{1}{\delta\left(A\right)}\max\quantity{\delta\left(A\cap
B_{k}\right)\mid k\in[m]}\leq\beta_{X}(A^{c},m)+\varepsilon,$

which implies that $\beta_{X}(A,m)\leq\beta_{X}(A^{c},m)$. Thus
$\beta(X,m)\leq\beta$ as claimed. ∎

Let $\mathcal{T}^{n}$ be the set of all non-singular linear transformations on
$\mathbb{R}^{n}$. The (multiplicative) Banach-Mazur metric
$d_{BM}^{M}:~{}\mathcal{K}^{n}\times\mathcal{K}^{n}\mapsto\mathbb{R}$ is
defined by

$d_{BM}^{M}(K_{1},K_{2})=\inf\quantity{\gamma\geq 1\mid\exists
T\in\mathcal{T}^{n},v\in\mathbb{R}^{n}\text{ s.t. }T(K_{2})\subset
K_{1}\subset\gamma T(K_{2})+v},~{}\forall K_{1},K_{2}\in\mathcal{K}^{n}.$

The infimum is clearly attained. When both $K_{1}$ and $K_{2}$ are symmetric
with respect to $o$, we have

$d_{BM}^{M}(K_{1},K_{2})=\inf\quantity{\gamma\geq 1\mid\exists
T\in\mathcal{T}^{n}\text{ s.t. }T(K_{2})\subset K_{1}\subset\gamma T(K_{2})}.$

In this situation, $d_{BM}^{M}(K_{1},K_{2})$ equals to the Banach-Mazur
distance between the Banach spaces $X$ and $Y$ having $K_{1}$ and $K_{2}$ as
unit balls, respectively. I.e.,

$d_{BM}^{M}(K_{1},K_{2})=d_{BM}^{M}(X,Y):=\inf\quantity{\left\|T\right\|\cdot\left\|T^{-1}\right\|\mid
T\text{ is an isomorphism from $X$ onto $Y$}}.$

We have the following result showing the stability of $\beta(X,m)$ with
respect to $X$ in the sense of Banach-Mazur metric.

######  Theorem 2.

If $X=(\mathbb{R}^{n},\left\|\cdot\right\|_{X})$ and
$Y=(\mathbb{R}^{n},\left\|\cdot\right\|_{Y})$ are two Banach space satisfying
$d_{BM}^{M}(X,Y)\leq\gamma$ for some $\gamma\geq 1$, then

$\beta(X,m)\leq\gamma\beta(Y,m),~{}\forall m\in\mathbb{Z}^{+}.$

###### Proof.

By applying a suitable linear transformation if necessary, we may assume that

$B_{Y}\subseteq B_{X}\subseteq\gamma B_{Y}.$

In this situation we have, for each $x\in\mathbb{R}^{n}$,

$\displaystyle\left\|x\right\|_{X}=\inf\quantity{\lambda>0\mid x\in\lambda
B_{X}}$ $\displaystyle\leq\inf\quantity{\lambda>0\mid x\in\lambda B_{Y}}$
$\displaystyle=\left\|x\right\|_{Y}$
$\displaystyle=\inf\quantity{\lambda\gamma>0\mid x\in\lambda\gamma B_{Y}}$
$\displaystyle=\gamma\inf\quantity{\lambda>0\mid x\in\lambda\gamma B_{Y}}$
$\displaystyle\leq\gamma\inf\quantity{\lambda>0\mid x\in\lambda
B_{X}}=\gamma\left\|x\right\|_{X}.$

Hence,

(3)
$\left\|x\right\|_{X}\leq\left\|x\right\|_{Y}\leq\gamma\left\|x\right\|_{X}.$

In the rest of this proof, we denote by $\delta_{X}(A)$ and $\delta_{Y}(A)$
the diameter of a bounded subset $A$ of $\mathbb{R}^{n}$ with respect to
$\left\|\cdot\right\|_{X}$ and $\left\|\cdot\right\|_{Y}$, respectively. By
(3), we have

(4) $\delta_{X}(A)\leq\delta_{Y}(A)\leq\gamma\delta_{X}(A),~{}\forall
A\in\mathcal{B}^{n}.$

Let $A$ be a bounded subset of $X$. Then $A$ is also bounded in $Y$. Let
$A^{c}$ be a completion of $A$ in $Y$. For any $\varepsilon>0$, there exists a
collection $\quantity{B_{i}\mid i\in[m]}$ of subsets of $A^{c}$ such that
$A^{c}$ is the union of this collection and that

$\frac{1}{\delta_{Y}(A^{c})}\max\quantity{\delta_{Y}(B_{i})\mid
i\in[m]}\leq\beta_{Y}(A^{c},m)+\varepsilon.$

Then

$\displaystyle A=A\cap A^{c}=\bigcup\limits_{i\in[m]}(B_{i}\cap A),$

and, by (4),

$\displaystyle\frac{1}{\delta_{X}(A)}\max\quantity{\delta_{X}(B_{i}\cap A)\mid
i\in[m]}$
$\displaystyle\leq\frac{\gamma}{\delta_{Y}(A)}\max\quantity{\delta_{Y}(B_{i})\mid
i\in[m]}$
$\displaystyle=\frac{\gamma}{\delta_{Y}\quantity(A^{c})}\max\quantity{\delta_{Y}(B_{i})\mid
i\in[m]}$ $\displaystyle\leq\gamma(\beta_{Y}(A^{c},m)+\varepsilon).$

Therefore, $\beta_{X}(A,m)\leq\gamma\beta_{Y}(A^{c},m)$. It follows that
$\beta(X,m)\leq\gamma\beta(Y,m)$. ∎

######  Corollary 3.

If $X=(\mathbb{R}^{n},\left\|\cdot\right\|_{X})$ and
$Y=(\mathbb{R}^{n},\left\|\cdot\right\|_{Y})$ are isometric, then
$\beta(X,m)=\beta(Y,m)$.

######  Proposition 4.

Let $l_{\infty}^{n}=(\mathbb{R}^{n},\left\|\cdot\right\|_{\infty})$. Then
$\beta(l_{\infty}^{n},2^{n})=\frac{1}{2}$.

###### Proof.

Put $X=l_{\infty}^{n}$. Then every complete set in $X$ is a homothetic copy of
$B_{X}$, see [11] and [27]. Therefore,

$\beta_{X}(A,2^{n})=\beta_{X}(B_{X},2^{n}),~{}\forall A\in\mathcal{C}_{X}.$

Thus it sufficies to show $\beta_{X}(B_{X},2^{n})=\frac{1}{2}$.

On the one hand, $B_{X}=\frac{1}{2}B_{X}+\frac{1}{2}V$, where $V$ is the set
of vertices of $B_{X}$. Since the cardinality of $V$ is $2^{n}$, we have
$\beta_{X}(B_{X},2^{n})\leq\frac{1}{2}$.

On the other hand, suppose that $B_{X}$ is the union of $2^{n}$ of its subsets
$B_{1},\cdots,B_{2^{n}}$. For each $i\in[2^{n}]$, let $B_{i}^{c}$ be a
completion of $B_{i}$. Then

$B_{X}\subseteq\bigcup\limits_{i\in[2^{n}]}B_{i}^{c}.$

It follows that

$\operatorname{vol}{B_{X}}\leq\sum\limits_{i\in[2^{n}]}\operatorname{vol}{B_{i}^{c}}\leq\sum\limits_{i\in[2^{n}]}\quantity(\frac{1}{2^{n}})\max\quantity{\delta_{X}(B_{i}^{c})\mid
i\in[2^{n}]}\operatorname{vol}{B_{X}},$

which implies that $\max\quantity{\delta_{X}(B_{i}^{c})\mid i\in[2^{n}]}\geq
1$. Thus $\beta_{X}(B_{X},2^{n})\geq\frac{1}{2}$, which completes the proof. ∎

######  Corollary 5.

Let $X=(\mathbb{R}^{n},\left\|\cdot\right\|)$. If
$d_{BM}^{M}(X,l_{\infty}^{n})<2$, then $\beta(X,2^{n})<1$.

We end this section with the following result.

######  Proposition 6.

$\sup\quantity{\beta(X,4)\mid\text{X is a two-dimensional Banach
space}}=\frac{\sqrt{2}}{2}$.

###### Proof.

Put $\eta=\sup\quantity{\beta(X,4)\mid X\text{ is a two-dimensional Banach
space}}$. Let $K\subset\mathbb{R}^{2}$ be a planar convex body. By the main
result in [23], $K$ can be covered by four translates of
$\frac{\sqrt{2}}{2}K$. It follows that $\beta_{X}(K,4)\leq\frac{\sqrt{2}}{2}$
holds for each two-dimensional Banach space $X$. Thus,
$\eta\leq\frac{\sqrt{2}}{2}$.

Let $X=l_{2}^{2}$ and $B_{X}$ be the unit disk of $l_{2}^{2}$. To show that
$\eta\geq\frac{\sqrt{2}}{2}$, we only need to prove
$\beta_{X}(B_{X},4)\geq\frac{\sqrt{2}}{2}$. Suppose the contrary that $B_{X}$
is the union of $A_{1},A_{2},A_{3},A_{4}$, where

$\max\quantity{\delta\left(A_{i}\right)\mid i\in[4]}<\frac{\sqrt{2}}{2}.$

Let $v_{1}$, $v_{2}$, $v_{3}$, and $v_{4}$ be the vertices of a square
inscribed in the unit circle $S_{X}$ of $X$. Then for any
$\quantity{i,j}\subset[4]$

$\left\|v_{i}-v_{j}\right\|\geq\sqrt{2}.$

Assume without loss of generality that $v_{1}\in A_{1}$ and $v_{2}\in A_{2},$
then

$\frac{v_{1}+v_{2}}{\left\|v_{1}+v_{2}\right\|}\notin A_{1}\cup A_{2}.$

Assume that $\frac{v_{1}+v_{2}}{\left\|v_{1}+v_{2}\right\|}\in A_{3}$. Then
$v_{3},v_{4}\notin A_{1}\cup A_{2}\cup A_{3}$, and $A_{4}$ cannot contain both
$v_{3}$ and $v_{4}$, a contradiction. Thus,
$\beta_{X}(B_{X},4)\geq\frac{\sqrt{2}}{2}$ as claimed. ∎

##  3\. Estimating $\beta(X,m)$ via $\beta_{X}(B_{X},m)$

Let $S$ be a simplex in $X=(\mathbb{R}^{n},\left\|\cdot\right\|)$. If the
distance between each pair of vertices of $S$ all equals to
$\delta\left(S\right)$, then we say that $S$ is _equilateral_.

######  Proposition 7.

Let $T$ be a triangle in $\mathbb{R}^{2}$ and
$X=(\mathbb{R}^{2},\left\|\cdot\right\|)$. Then
$\beta_{X}(T,4)\leq\frac{1}{2}$. If $T$ is equilateral in $X$, then
$\beta_{X}(T,4)=\frac{1}{2}$.

###### Proof.

We only need to consider the case when $T$ is equilateral. Assume without loss
of generality that $\delta\left(T\right)=1$. It is clear that
$\beta_{X}(T,4)\leq\frac{1}{2}$. Denote by $\quantity{a,b,c}$ the set of
vertices of $T$, and by $\quantity{p,q,r}$ the set of midpoints of three sides
of $T$. Then
$\left\|p-q\right\|=\left\|p-r\right\|=\left\|q-r\right\|=\frac{1}{2}$.

Suppose the contrary that $\beta_{X}(T,4)<\frac{1}{2}$. Then there exist four
subsets $T_{1},T_{2},T_{3},T_{4}$ of $T$ such that $T=\bigcup_{i\in[4]}T_{i}$
and that $\max\quantity{\delta\left(T_{i}\right)\mid i\in[4]}<\frac{1}{2}$. We
may assume that $a\in T_{1}$, $b\in T_{2}$, and $c\in T_{3}$. Since
$\quantity{p,q,r}\cap\bigcup_{i\in[3]}T_{i}=\emptyset$, we have
$\quantity{p,q,r}\subseteq T_{4}$, which is impossible. Thus
$\beta_{X}(T,4)=\frac{1}{2}$ as claimed. ∎

######  Proposition 8.

Let $T$ be a simplex in $X=(\mathbb{R}^{3},\left\|\cdot\right\|)$. Then

$\beta_{X}(T,8)\leq\frac{9}{16}.$

###### Proof.

Denote by $\quantity{v_{i}\mid i\in[4]}$ the set of vertices of $T$. Without
loss of generality we may assume that $o=\frac{1}{4}\sum_{i\in[4]}v_{i}$. For
each $i\in[4]$, put $T_{i}=\frac{7}{16}v_{i}+\frac{9}{16}T$. Then the portion
of $T$ not covered by $\bigcup_{i\in[4]}T_{i}$ is

$T_{5}=\quantity{\sum_{i\in[4]}\lambda_{i}v_{i}\mid\lambda_{i}\in\quantity[0,\frac{7}{16}],~{}\forall
i\in[4],~{}\sum_{i\in[4]}\lambda_{i}=1}.$

Suppose that $\sum_{i\in[4]}\lambda_{i}v_{i}\in T_{5}$. Then

$\displaystyle\frac{4}{7}\sum_{i\in[4]}\lambda_{i}v_{i}-o$
$\displaystyle=\frac{4}{7}\sum_{i\in[4]}\lambda_{i}v_{i}-\frac{1}{4}\sum_{i\in[4]}v_{i}$
$\displaystyle=-\sum_{i\in[4]}\quantity(\frac{1}{4}-\frac{4}{7}\lambda_{i})v_{i}$
$\displaystyle=-\quantity(\sum_{j\in[4]}\quantity(\frac{1}{4}-\frac{4}{7}\lambda_{j}))\sum_{i\in[4]}\frac{\frac{1}{4}-\frac{4}{7}\lambda_{i}}{\sum_{j\in[4]}(\frac{1}{4}-\frac{4}{7}\lambda_{j})}v_{i}$
$\displaystyle\in-\frac{3}{7}T.$

It follows that $\sum_{i\in[4]}\lambda_{i}v_{i}\in-\frac{3}{4}T$. Thus
$T_{5}\subset-\frac{3}{4}T$. It is not difficult to verify that $T$ can be
covered by $4$ translates of $\frac{3}{4}T$, which implies that $T_{5}$ can be
covered by $4$ translates of $-\frac{9}{16}T$. Therefore,
$\beta_{X}(T,8)\leq\frac{9}{16}$. ∎

######  Proposition 9.

Let $T$ be a simplex in $X=(\mathbb{R}^{3},\left\|\cdot\right\|)$. Then

$\beta_{X}(T,9)\leq\frac{9}{17}.$

###### Proof.

We use the idea in the proof of Proposition 8. For each $i\in[4]$, put
$T_{i}=\frac{8}{17}v_{i}+\frac{9}{17}T$. Then the portion of $T$ not covered
by $\bigcup_{i\in[4]}T_{i}$ is

$T_{5}=\quantity{\sum_{i\in[4]}\lambda_{i}v_{i}\mid\lambda_{i}\in\quantity[0,\frac{8}{17}],~{}\forall
i\in[4],~{}\sum_{i\in[4]}\lambda_{i}=1}.$

As in the proof of Proposition 8, $T_{5}\subset-\frac{15}{17}T$. By using the
idea in the proof of Proposition 8 again, one can show that
$\beta_{X}(T,5)\leq\frac{3}{5}$. Therefore, $T_{5}$ is the union of $5$
subsets of $T_{5}$ whose diameters are not larger than
$\frac{9}{17}\delta\left(T\right)$. It follows that
$\beta_{X}(T,9)\leq\frac{9}{17}$. ∎

######  Remark 10.

The estimations in Proposition 8 and Proposition 9 are independent of the
choice of norm on $\mathbb{R}^{3}$.

For a convex body $K$, _the Minkowski measure of symmetry_ , denoted by
$s(K)$, is defined as

$s(K)=\min\quantity{\lambda>0\mid\exists x\in X\text{ s.t. }K+x\subset-\lambda
K}.$

It is known that

$1\leq s(K)\leq n,~{}\forall K\in\mathcal{K}^{n};$

the equality on the left holds if and only if $K$ is centrally symmetric, and
the equality on the right holds if and only if $K$ is a simplex (cf. [29]).

The following lemma shows the stability of $\beta_{X}(K,m)$ with respect to
$K$.

######  Lemma 11.

Let $X=(\mathbb{R}^{n},\left\|\cdot\right\|)$, and $K$ and $L$ be two convex
bodies in $X$. If there exist a number $\gamma\geq 1$ and a point
$c\in\mathbb{R}^{n}$ such that

$K\subseteq L\subseteq\gamma K+c,$

then, for each $m\in\mathbb{Z}^{+}$, we have

$\beta_{X}(L,m)\leq\gamma\beta_{X}(K,m).$

###### Proof.

For each $\varepsilon>0$, there exists a collection $\quantity{K_{i}\mid
i\in[m]}$ of subsets of $\gamma K+c$ such that

$\gamma K+c=\bigcup_{i\in[m]}K_{i}$

and

$\delta\left(K_{i}\right)\leq\gamma\delta\left(K\right)\beta_{X}(K,m)+\varepsilon\leq\gamma\delta\left(L\right)\beta_{X}(K,m)+\varepsilon,~{}\forall
i\in[m].$

Since

$L=L\cap(\gamma K+c)=\bigcup_{i\in[m]}\quantity(L\cap K_{i}),$

we have

$\beta_{X}(L,m)\leq\gamma\beta_{X}(K,m)+\frac{\varepsilon}{\delta\left(L\right)}.$

Since $\varepsilon$ is arbitrary, $\beta_{X}(L,m)\leq\gamma\beta_{X}(K,m)$ as
claimed. ∎

######  Theorem 12.

Let $X=(\mathbb{R}^{3},\left\|\cdot\right\|)$, $m\in\mathbb{Z}^{+}$, and

$\eta=\sup\quantity{\beta_{X}(T,m)\mid T\text{ is a simplex in
$\mathbb{R}^{3}$}}.$

We have

$\beta(X,m)\leq\min\limits_{\varepsilon\in(0,1/3)}\max\quantity{\quantity(1+\frac{4\varepsilon}{1-3\varepsilon})\eta,\frac{2(3-\varepsilon)}{4-\varepsilon}\beta_{X}(B_{X},m)}.$

###### Proof.

Let $K$ be a complete set in $X$, $\varepsilon$ be a number in
$\quantity(0,\frac{1}{3})$. We distinguish two cases.

Case 1. The Banach-Mazur distance from $K$ to three-dimensional simplices is
bounded from the above by

$1+\frac{4\varepsilon}{1-3\varepsilon}.$

Then there exist a tetrahedron $T$ and a point $c\in\mathbb{R}^{3}$ such that

$T\subset K\subset\quantity(1+\frac{4\varepsilon}{1-3\varepsilon})T+c.$

By Lemma 11, we have

$\beta_{X}(K,m)\leq\quantity(1+\frac{4\varepsilon}{1-3\varepsilon})\eta.$

Case 2. The Banach-Mazur distance from $K$ to three-dimensional simplex is at
least

$1+\frac{4\varepsilon}{1-3\varepsilon}.$

From Theorem 2.1 in [25], it follows that

$s(K)\leq 3-\varepsilon.$

Denote by $R(K)$ the circumradius of $K$. Theorem 1.1 in [7] shows that

$s(K)=\frac{R(K)/\delta\left(K\right)}{1-R(K)/\delta\left(K\right)}.$

It follows that

$\frac{R(K)}{\delta\left(K\right)}\leq\frac{3-\varepsilon}{4-\varepsilon}.$

By a suitable translation if necessary, we may assume that

$K\subseteq\frac{3-\varepsilon}{4-\varepsilon}\delta\left(K\right)B_{X}.$

For each $\gamma>0$, there exists a collection $\quantity{B_{i}\mid i\in[8]}$
such that

$B_{X}=\bigcup_{i\in[8]}B_{i}\mbox{\quad and\quad}\delta\left(B_{i}\right)\leq
2\beta_{X}(B_{X},m)+\gamma,~{}\forall i\in[m].$

It follows that

$\beta_{X}(K,m)\leq\frac{2(3-\varepsilon)}{4-\varepsilon}\beta_{X}(B_{X},m)+\frac{3-\varepsilon}{4-\varepsilon}\gamma.$

Hence

$\beta_{X}(K,m)\leq\frac{2(3-\varepsilon)}{4-\varepsilon}\beta_{X}(B_{X},m).$

This completes the proof. ∎

######  Corollary 13.

Let $X=(\mathbb{R}^{3},\left\|\cdot\right\|)$. If
$\beta_{X}(B_{X},8)<\frac{221}{328}$, then $\beta(X,8)<1$.

###### Proof.

Since $\frac{2(3-\frac{7}{57})}{4-\frac{7}{57}}\frac{221}{328}=1$ and

$\frac{2(3-\varepsilon)}{4-\varepsilon}=2-\frac{2}{4-\varepsilon}$

is continuous with respect to $\varepsilon$ on $(0,\frac{1}{3})$, there exists
a number $\varepsilon_{0}<\frac{7}{57}$ such that

$\frac{2(3-\varepsilon_{0})}{4-\varepsilon_{0}}\beta_{X}(B_{X},8)<1$

It follows that

$\beta(X,8)\leq\max\quantity{\quantity(1+\frac{4\varepsilon_{0}}{1-3\varepsilon_{0}})\frac{9}{16},\frac{2(3-\varepsilon_{0})}{4-\varepsilon_{0}}\beta_{X}(B_{X},8)}<1.\qed$

In particular, Corollary 13 shows that $\beta(l_{1}^{3},8)<1$ since the unit
ball of $l_{1}^{3}$ can be covered by $8$ balls having radius
$\frac{2}{3}<\frac{221}{328}$. By solving the optimization problem

$\min_{\varepsilon\in(0,1/3)}\max\quantity{\quantity(1+\frac{4\varepsilon}{1-3\varepsilon})\frac{9}{16},\frac{2(3-\varepsilon)}{4-\varepsilon}\frac{2}{3}},$

one can show that $\beta(l_{1}^{3},8)\leq 0.989\ldots$. This estimation can be
improved, see the next section.

##  4\. $\beta\quantity(l_{p}^{3},8)$

######  Lemma 14.

For each $p\in[1,2]$,
$d_{BM}^{M}\quantity(l_{p}^{3},l_{\infty}^{3})\leq\frac{\sqrt{18\cdot
19}}{10}\approx 1.85$.

###### Proof.

Put $c_{1}=(3,3,-2)$, $c_{2}=(-2,3,3)$, $c_{3}=(3,-2,3)$. Denote by $Q$ the
parallelipiped having

$\quantity{\sum\limits_{i\in[3]}\sigma_{i}c_{i}\mid\sigma_{i}\in\quantity{-1,1},~{}\forall
i\in[3]}$

as the set of vertices. We have

$\displaystyle\max\quantity{\left\|\sum\limits_{i\in[3]}\sigma_{i}c_{i}\right\|_{p}\mid
i\in[3]}$
$\displaystyle=\max\quantity{\left\|(4,4,4)\right\|_{p},\left\|(-2,8,-2)\right\|_{p},\left\|(8,-2,-2)\right\|_{p},\left\|(2,2,-8)\right\|_{p}}$
$\displaystyle=\max\quantity{\left\|(4,4,4)\right\|_{p},\left\|(-2,8,-2)\right\|_{p}}$
$\displaystyle=\left\|(-2,8,-2)\right\|_{p}=2\left\|(-1,4,-1)\right\|_{p}=2\left\|(1,1,4)\right\|_{p}.$

It follows that

$\frac{1}{2\left\|(1,1,4)\right\|_{p}}Q\subset B_{p}^{3},$

where $B_{p}^{3}$ is the unit ball of $l_{p}^{3}$. Let $q$ be the number
satisfying

$\frac{1}{p}+\frac{1}{q}=1,$

and let $f_{1}$, $f_{2}$, $f_{3}$ be linear functionals defined on $l_{p}^{3}$
such that, for any $(\alpha,\beta,\gamma)\in\mathbb{R}^{3}$,

$\displaystyle
f_{1}((\alpha,\beta,\gamma))=\frac{1}{100}\quantity(15\alpha-5\beta+15\gamma),$
$\displaystyle
f_{2}((\alpha,\beta,\gamma))=\frac{1}{100}\quantity(-5\alpha+15\beta+15\gamma),$
$\displaystyle
f_{3}((\alpha,\beta,\gamma))=\frac{1}{100}\quantity(15\alpha+15\beta-5\gamma).$

Then

$\displaystyle
c_{3}+\operatorname{span}\quantity{c_{1},c_{2}}=\quantity{x\in\mathbb{R}^{3}\mid
f_{1}(x)=1},$ $\displaystyle
c_{2}+\operatorname{span}\quantity{c_{1},c_{3}}=\quantity{x\in\mathbb{R}^{3}\mid
f_{2}(x)=1},$ $\displaystyle
c_{1}+\operatorname{span}\quantity{c_{2},c_{3}}=\quantity{x\in\mathbb{R}^{3}\mid
f_{3}(x)=1}.$

Thus the distances from the origin $o$ to the facets of $Q$ all equals to

$\frac{100}{\left\|(15,-5,15)\right\|_{q}}.$

It follows that

$\frac{1}{2\left\|(1,1,4)\right\|_{p}}Q\subset
B_{p}^{3}\subset\frac{\left\|(15,-5,15)\right\|_{q}}{100}Q=\frac{\left\|(1,1,4)\right\|_{p}\left\|(3,1,3)\right\|_{q}}{10}\frac{1}{2\left\|(1,1,4)\right\|_{p}}Q,$

which implies that

$d_{BM}^{M}(l_{p}^{3},l_{\infty}^{3})\leq\frac{\left\|(1,1,4)\right\|_{p}\left\|(3,1,3)\right\|_{q}}{10}\leq\frac{\left\|(1,1,4)\right\|_{2}\left\|(3,1,3)\right\|_{2}}{10}=\frac{\sqrt{18\cdot
19}}{10}.\qed$

######  Remark 15.

The last inequality in Lemma 14 can be verified in the following way. Put

$f(p)=\quantity(4^{p}+2)^{\frac{1}{p}}\cdot\quantity(2\cdot
3^{\frac{p}{p-1}}+1)^{\frac{p-1}{p}}.$

Numerical results show that $f^{\prime}(p)=0$ has a unique solution
$p_{0}\approx 1.320$ in $[1,2]$, and

$f(p_{0})\approx 17.550<f(2).$

Moreover, $f(1)<f(2)$. Thus $f(p)$ is maximized at $p=2$.

Numerical results show that when $p\in[1,1.736)$,

$\frac{\left\|(1,1,4)\right\|_{p}\left\|(3,1,3)\right\|_{q}}{10}\leq\frac{9}{5}.$

The estimation in Lemma 14 could be improved by choosing points $c_{1}$,
$c_{2}$, $c_{3}$ more carefully for different $p\in[1,2]$.

######  Theorem 16.

We have the following estimation:

$\beta\quantity(l_{p}^{3},8)\leq\begin{cases}\frac{\sqrt{18\cdot
19}}{20},&p\in[1,2),\\\ \frac{3^{1/p}}{2},&p\in[2,+\infty].\end{cases}$

###### Proof.

First we consider the case when $p\in[2,+\infty]$. By Proposition 37.6 in
[28], $d_{BM}^{M}(l_{p}^{3},l_{\infty}^{3})=3^{1/p}$, this together with
Theorem 2, implies that
$\beta(l_{p}^{3},8)\leq\frac{3^{1/p}}{2}\leq\frac{\sqrt{3}}{2}$.

The case when $p\in[1,2]$ follows directly from Lemma 14 and Theorem 2. ∎

##  5\. Acknowledgements

This work is supported by the National Natural Science Foundation of China
(grant numbers: 11921001 and 12071444), the National Key Research and
Development Program of China (2018YFA0704701), and the Natural Science
Foundation of Shanxi Province of China (201901D111141). The authors are
grateful to Professor C. Zong for his supervision and discussion.

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