# DensityTool: A post-processing tool for space- and spin-resolved density of
states from VASP

Lucas Lodeiro  Tomáš Rauch  Departamento de Química, Facultad de Ciencias,
Universidad de Chile, Las Palmeras 3425, Ñuñoa 7800003, Santiago, Chile
Institut für Festkörpertheorie und -Optik, FSU Jena, Max-Wien-Platz 1, 07743
Jena, Germany, Germany

###### Abstract

The knowledge of the local electronic structure of heterogeneous solid
materials is crucial for understanding their electronic, magnetic, transport,
optical, and other properties. VASP, one of the mostly used packages for
density-functional calculations, provides local electronic structure either by
projecting the electronic wave functions on atomic spheres, or as a band-
decomposed partial charge density. Here, we present a simple tool which takes
the partial charge density and the energy eigenvalues calculated by VASP as
input and constructs local charge and spin densities. The new data provides a
much better spatial resolution than the projection on the atomic spheres. It
can be visualized directly in the real space e.g. with Vesta, or averaged
along planes spanned by two of the lattice vectors of the periodic unit cell.
The plane-averaged local (spin) density of states can be easily plotted e.g.
as color-coded data using almost any plotting program. DensityTool can be
applied to manipulate, visualize, and understand the local electronic
structure of any system calculated with VASP. We expect it to be useful
especially for researchers concerned with inhomogeneous systems, such as
interfaces, defects, surfaces, adsorbed molecules, or hybrid inorganic-organic
composites.

###### keywords:

FORTRAN ; Electronic Structure; VASP; Local Density Of States

††journal: Computer Physics Communications

PROGRAM SUMMARY

Program Title: DensityTool
CPC Library link to program files: (to be added by Technical Editor)
Developer’s repository link: https://github.com/llodeiro/DensityTool
Code Ocean capsule: (to be added by Technical Editor)
Licensing provisions: MIT
Programming language: FORTRAN
Supplementary material: A complementary User Manual and examples can be found
in the developer’s repository link.
Nature of problem(approx. 50-250 words): Total and local density of states
(LDOS) are widely-used quantities for understanding the electronic structure
of condensed matter. In VASP, one of the mostly used packages for numerically
solving the Kohn-Sham equations of density-functional theory, one does not
directly obtain the LDOS. Usually, one works with partial density of states
(PDOS), where the wave function is projected onto atomic orbitals.
Alternatively, it is possible to calculate the band- and wavevector-dependent
charge densities. Together with the band structure, LDOS can be obtained using
post-processing tools.

Solution method(approx. 50-250 words): DensityTool combines the band structure
with band- and wavevector-dependent charge densities calculated by VASP to
construct LDOS. In addition, spin-resolved LDOS (LSDOS) can be calculated for
magnetic systems. Thus obtained LDOS and LSDOS can be further processed with
DensityTool. In particular, it can be averaged along chosen spatial
directions, which is convenient for example to demonstrate the spatially-
resolved electronic structure of inhomogeneous systems.
DensityTool can handle all three types of approaches implemented in VASP (non-
magnetic, collinear magnetic, non-collinear magnetic including spin-orbit
coupling), as well as all types of self-consistent band calculations as DFT
(LDA, GGA, mGGA and hybrid functionals) and HF.
For convenience, the output of DensityTool follows the format of VASP (the
CHGCAR file), which can be easily used in other programs for visualization. We
provide a sample python script to plot averaged L(S)DOS from the calculated
data.

Additional comments including restrictions and unusual features (approx.
50-250 words):

##  1 Introduction

Density-functional theory (DFT) [1, 2] is an extremely successful approach to
calculating ground-state electronic properties of condensed matter in the one-
particle approximation [3]. In this work we will refer to applications of DFT
to periodic systems, which allows to use the implications of the Bloch theorem
[4]. The systems are described in terms of periodic unit cells, which may be
also of artificial nature. This allows to describe with this approach also
non-periodic materials like finite slabs with surfaces, 2D materials, or
molecules.

For a given periodic system, the Kohn-Sham [1, 2] equations are solved with
the wave functions $\varphi_{n,{\bf k}}({\bf r})$ and their eigenvalues
$\epsilon_{n,{\bf k}}$ as a solution. The square of the occupied wave
functions yields the partial charge density $P_{n,{\bf k}}({\bf
r})=\left|\varphi_{n,{\bf k}}({\bf r})\right|^{2}$ and the eigenvalues are
interpreted as the one-particle energy levels.

Using the energy levels, it is common to define the density of states (DOS) as

$D(E)=\frac{N_{e}}{(2\pi)^{3}}\sum_{n}\int_{\mathrm{BZ}}\delta(E-\epsilon_{n,{\bf
k}})\ d^{3}k$ (1)

with $N_{e}$ the band occupancy. The DOS is an important property which
describes the complete electronic structure of the entire material. This is
very convenient in case of periodic solids with rather smaller unit cells,
where the spatial variation of DOS is not important.

On the other hand, when one is studying the electronic structure of
inhomogeneous systems like e.g. interfaces, defects, surfaces, adsorbed
molecules, or hybrid inorganic-organic composites, access to spatially
resolved DOS is more convenient. Such a quantity is the local DOS (LDOS) which
combines the energy levels with the partial charge density:

$L(E,{\bf
r})=\frac{N_{e}}{(2\pi)^{3}}\sum_{n}\int_{\mathrm{BZ}}\delta(E-\epsilon_{n,{\bf
k}})P_{n,{\bf k}}({\bf r})\ d^{3}k.$ (2)

Alternatively, LDOS can be understood as energy-resolved partial charge
density, starting with the definition of the total charge density

$\rho({\bf
r})=\frac{N_{e}{\Omega_{\mathrm{cell}}}}{(2\pi)^{3}}\sum_{n}\int_{\mathrm{BZ}}f_{n,{\bf
k}}P_{n,{\bf k}}({\bf r})\ d^{3}k$ (3)

with the unit cell volume ${\Omega_{\mathrm{cell}}}$ and the Fermi-Dirac
distribution function $f_{n,{\bf k}}$.

In magnetic systems the spin becomes an additional quantum number when it is
conserved (e.g., for collinear magnetism and systems without spin-orbit
coupling). We thus have the spin-resolved quantities

$P^{\updownarrows}_{n,{\bf k}}({\bf r})=\left|\varphi^{\updownarrows}_{n,{\bf
k}}({\bf r})\right|^{2},$ (4) $\rho^{\updownarrows}({\bf
r})=\frac{{\Omega_{\mathrm{cell}}}}{(2\pi)^{3}}\sum_{n}\int_{\mathrm{BZ}}f^{\updownarrows}_{n,{\bf
k}}P^{\updownarrows}_{n,{\bf k}}({\bf r})\ d^{3}k,$ (5)
$D^{\updownarrows}(E)=\frac{1}{(2\pi)^{3}}\sum_{n}\int_{\mathrm{BZ}}\delta(E-\epsilon^{\updownarrows}_{n,{\bf
k}})\ d^{3}k,$ (6)

and

$L^{\updownarrows}(E,{\bf
r})=\frac{1}{(2\pi)^{3}}\sum_{n}\int_{\mathrm{BZ}}\delta(E-\epsilon^{\updownarrows}_{n,{\bf
k}})P^{\updownarrows}_{n,{\bf k}}({\bf r})\ d^{3}k.$ (7)

Using these, the total quantities can be recovered as the sum of the spin-
resolved parts, e.g., the total charge density becomes

$\rho({\bf r})=\rho^{\uparrow}({\bf r})+\rho^{\downarrow}({\bf r}).$ (8)

In addition, information about the spin-part of the magnetization can be
deduced from the spin density

$s({\bf r})=\rho^{\uparrow}({\bf r})-\rho^{\downarrow}({\bf r})$ (9)

and its spatially-resolved generalization, the local spin DOS (LSDOS)

$S(E,{\bf
r})=\frac{1}{(2\pi)^{3}}\sum_{n}\int_{\mathrm{BZ}}\left[\delta(E-\epsilon^{\uparrow}_{n,{\bf
k}})P^{\uparrow}_{n,{\bf k}}({\bf r})-\delta(E-\epsilon^{\downarrow}_{n,{\bf
k}})P^{\downarrow}_{n,{\bf k}}({\bf r})\right]d^{3}k.$ (10)

LSDOS can be used to study the energy- and position-dependent contributions to
the spin magnetization. For example, such an approach can be applied to
identify the origin of local magnetization at surfaces [5, 6] or defects [7,
8].

In the following we present a tool which takes $P_{n,{\bf k}}$ and
$\epsilon_{n,{\bf k}}$ calculated with the VASP [9, 10, 11] package as input
and calculates $L(E,{\bf r})$ and $S(E,{\bf r})$. Furthermore, the
functionality of the tool allows to calculate $L$ and $S$ averaged along some
direction for a better interpretation of the data.

##  2 Methodology

After successfully finishing a self-consistent calculation with the VASP
package, the eigenstates $\epsilon_{n,{\bf k}}$ are listed in the EIGENVAL
file. Setting further LPARD=T, LSEPB=T, LSEPK=T in the input file (INCAR), the
partial charge densities $P_{n,{\bf k}}$ are calculated by VASP from the
preconverged wave functions and stored in the PARCHG.nnnn.kkkk files. These
files are the basic input of DensityTool.

In VASP, the structure of the calculated system is defined by a periodic unit
cell spanned by the three lattice vectors ${\bf L}_{i}$, $i=1,2,3$. The volume
of the unit cell
$\Omega_{\mathrm{cell}}=L_{1}L_{2}L_{3}|\hat{L}_{3}\cdot(\hat{L}_{1}\times\hat{L}_{2})|$
with $L_{i}$ and $\hat{L}_{i}$ the magnitude and the unit vector of the
lattice vectors ${\bf L}_{i}$. In the numerical calculation, the partial
charge density (along with other quantities) is sampled on a discrete mesh in
the real space with $N_{i}$ mesh points along each of the three lattice
vectors and the eigenvalues (band structure) are sampled on another discrete
mesh in the Brillouin zone in the reciprocal space. The spacing of both meshes
can be set in VASP input files. Therefore, the L(S)DOS is also expressed on
the real-space discrete mesh. To evaluate the Brillouin-zone integrals above,
their discretized version

$\frac{1}{(2\pi)^{3}}\int_{\mathrm{BZ}}\ldots\
d^{3}k\qquad\longrightarrow\qquad{}\frac{1}{\Omega_{\mathrm{cell}}}\sum_{{\bf
k}\ \in\ \mathrm{BZ}}\ldots W_{{\bf k}}$ (11)

must be employed. $W_{\bf k}$ are weights of the ${\bf k}$-points and their
values depend on the discretization of the reciprocal unit cell in the
original VASP calculation.

With DensityTool we are aiming at the construction of the L(S)DOS and its
visualization. Therefore, it is useful to express the $E$-dependence of $L$
and $S$ on a regular mesh, as it is already the case for the spatial
coordinates. For this purpose, we approximate $\delta(E-\epsilon_{n,{\bf k}})$
in all equations by

$\delta(E-\epsilon_{n,{\bf k}})\longrightarrow g_{n,{\bf
k}}=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(E-\epsilon_{n,{\bf
k}})^{2}}{2\sigma^{2}}}$ (12)

with a proper smearing parameter $\sigma$. By this approximation $L(E,{\bf
r})$ and $S(E,{\bf r})$ becomes continuous in $E$ and in particular it can be
evaluated on an equidistant set of $E$ values. We further introduce a
threshold parameter $\eta$ and redefine Eq. (12) as

$\tilde{g}_{n,{\bf k}}(E,\sigma)=\left\\{\begin{array}[]{cl}g_{n,{\bf
k}}(E,\sigma)&\text{for }g_{n,{\bf
k}}(E,\sigma)>\frac{\eta}{\sigma\sqrt{2\pi}}\\\ 0&\text{for }g_{n,{\bf
k}}(E,\sigma)\leq\frac{\eta}{\sigma\sqrt{2\pi}}.\\\ \end{array}\right.$ (13)

Thus, for a given energy $E$, only those $P_{n,{\bf k}}$ are considered, for
which $g_{n,{\bf k}}(E,\sigma)>\frac{\eta}{\sigma\sqrt{2\pi}}$. This procedure
ensures a substantial speed up of the summation over all $n$ and ${\bf k}$. As
an optimal compromise we propose to set $\eta=0.001$, for which $99.98\%$ of
the distribution area of $g_{n,{\bf k}}$ is recovered.

Finally, DensityTool can also calculate the L(S)DOS averaged on a plane
spanned by two of the lattice vectors for each of the discrete mesh points
along the third lattice vector and for each $E$. This way three-dimensional
data is obtained which can be visualized e.g. as a scatter plot with color
coded $z$-coordinate of the plot. For example, we choose to average the data
on the plane spanned by ${\bf L}_{1}$ and ${\bf L}_{2}$ with area
$A_{12}=L_{1}L_{2}|\hat{L}_{1}\times\hat{L}_{2}|$. We can thus define the
plane-averaged LDOS as

$\bar{L}(E,r_{3})=\frac{1}{A_{12}}\iint L(E,{\bf
r})|\hat{L}_{1}\times\hat{L}_{2}|dr_{1}dr_{2}.$ (14)

For our numerical evaluation it is more convenient to rewrite Eq. (14) as

$\bar{L}(E,r_{3})=\frac{N_{e}}{(2\pi)^{3}}\sum_{n}\int_{\mathrm{BZ}}\delta(E-\epsilon_{n,{\bf
k}})\bar{P}_{n,{\bf k}}({\bf r})\ d^{3}k,$ (15)

where $\bar{P}_{n,{\bf k}}(r_{3})$ is the partial charge average (PCA) over
the area $A_{12}$,

$\bar{P}_{n,{\bf k}}(r_{3})=\frac{1}{A_{12}}\iint P_{n,{\bf k}}({\bf
r})|\hat{L}_{1}\times\hat{L}_{2}|dr_{1}dr_{2}=\frac{1}{L_{1}L_{2}}\iint
P_{n,{\bf k}}({\bf r})dr_{1}dr_{2}.$ (16)

Note that the real-space integrals must be evaluated on the discrete mesh and
thus

$\frac{1}{A_{12}}\iint\ldots|\hat{L}_{1}\times\hat{L}_{2}|dr_{1}dr_{2}\qquad\longrightarrow\qquad{}\frac{1}{N_{1}N_{2}}\sum_{r_{1},r_{2}}\ldots.$
(17)

Using Eq. (16) plane-averaged LSDOS or total spin or charge density can be
calculated in analogy to the plane-averaged LDOS.

In this section we demonstrated how plane-averaged and total L(S)DOS is
calculated from the output of VASP with DensityTool. A final subtlety remains
related to magnetic systems with conserved (collinear) spin (ISPIN=2 in VASP).
In this case, the primary output of VASP is $P_{n,{\bf k}}({\bf r})$ and
$s_{n,{\bf k}}({\bf r})=P^{\uparrow}_{n,{\bf k}}({\bf
r})-P^{\downarrow}_{n,{\bf k}}({\bf r})$. The spin-resolved partial charge
density $P^{\updownarrows}_{n,{\bf k}}({\bf r})$ can be obtained from this
data as $P^{\updownarrows}_{n,{\bf k}}({\bf r})=\frac{1}{2}\left(P_{n,{\bf
k}}({\bf r})\pm s_{n,{\bf k}}({\bf r})\right)$. In addition, we provide an
alternative approach, which is computationally faster, but only applicable to
cases with small spin splitting of the energy levels. In this approximation we
use Eq. (2) to calculate LDOS, but we have to approximate $\epsilon_{n,{\bf
k}}\approx\frac{1}{2}(\epsilon^{\uparrow}_{n,{\bf
k}}+\epsilon^{\downarrow}_{n,{\bf k}})$. LSDOS can be approximately calculated
using the same approach, but by substituting $P_{n,{\bf k}}({\bf r})$ for
$s_{n,{\bf k}}({\bf r})$ in Eq. (2).

##  3 Software structure and workflow

DensityTool is written in FORTRAN. The code is provided at
https://github.com/llodeiro/DensityTool, including installation and execution
instructions. It is a post-processor to VASP and it takes its output written
to the EIGENVAL, CHGCAR, and PARCHG files as input. In addition, input
parameters are necessary, which control the exact routines that should be used
and the numerics. Most convenient way to do this is to execute DensityTool by
letting it read the input parameters from a text file (see DENSITYTOOL.IN in
our examples). Please find the list of the available input parameters in the
User Manual.

Depending on the choice of routines, different tasks can be performed by
DensityTool. For plane-averaged L(S)DOS, the PCA or partial spin average (PSA)
must be calculated first with Eq. (16). For each state characterized by
$(n,{\bf k})$, the plane-averaged quantity is written to an individual file.
This enables to conduct further calculations with much smaller files than the
original ones written by VASP. The main tasks are then to calculate the plane-
averaged L(S)DOS from the previously calculated PCA or PSA via Eq. (15). In
case of magnetic materials, it is possible to choose between the exact
treatment of the spin-split energy levels and the approximation described in
Sec. 2. Alternatively, it is also possible to calculate the full L(S)DOS
without performing the average (Eqs. (2) and (10)). Again, in case of magnetic
materials the user can choose to apply the approximation to the eigen-
energies. Finally, the plane-averaged total charge or spin density can be
calculated directly from the CHGCAR file written by VASP.

We summarize the available routines in Tab. 1 and provide their detailed
description in the User Manual.

|  nonmagnetic or magnetic with approximated energies |  magnetic with exact energies
---|---|---
| partial charge | partial spin | partial charge | partial spin
partial average | PCA | PSA | PARCHGSPIN
plane-averaged local density | LDOS | LSDOS | LDOSMAG | LSDOSMAG
full local density | LDOSFULL | LSDOSFULL | LDOSFULLMAG | LSDOSFULLMAG
total density | CHGCARAVG | CHGCARSPIN
Table 1: Routines available in DensityTool. The two columns refer to the use
of the approximation to the spin-split energy levels for magnetic materials
with conserved spin.

##  4 Results

We demonstrate the capabilities of DensityTool on the example of concrete
inhomogeneous systems. This type of systems can be seen as a prototypical
field where the tool can be successfully applied. We provide additional simple
examples at the code repository https://github.com/llodeiro/DensityTool. To
keep the amount of data low for these simple examples, the VASP calculations
were not converged with respect to the number of ${\bf k}$-points in these
examples, making the results unphysical and only suitable for testing.
Therefore, we do not show these results here. Note that there are also
published results obtained with DensityTool in its development version [12,
13].

In particular, we present our results calculated with DensityTool for two
inhomogeneous systems: a composite inorganic-organic system, and a perovskite
slab model.

###  4.1 Composite inorganic-organic system:

The composite inorganic-organic system of interest is a hydrogenized Si(111)
slab with the F6-TCNNQ molecule adsorbed to one of the surfaces. The atomic
structure of the calculated system is shown in the bottom panel of Fig. 1.
Find the details of the VASP electronic structure calculations in Ref. [12]
and more information about the system in Ref. [14].

Figure 1: LDOS (top), LSDOS (center), and the atomic structure (bottom) of the
hydrogenized Si(111) slab with the F6-TCNNQ molecule adsorbed to its surface.
Some of the data shown here was published previously in Ref. [12]. The figures
were created with Matplotlib v2.2.2, VESTA v3.4.6 [15], and Inkscape 0.92.

Using DensityTool, we calculated the LDOS and LSDOS of the system. We chose
the exact treatment of the spin-resolved energies. Thus, starting from the
partial charge density written by VASP, it was averaged in the plane parallel
to the Si surface for both spin orientations separately with the PARCHGSPIN
routine. Subsequently, the plane-averaged LDOS and LSDOS were calculated with
the LDOSMAG and LSDOSMAG routines, respectively. We present the results in the
top (LDOS) and central (LSDOS) panel of Fig. 1. It can be clearly seen that
the plane-averaged LDOS has particularly large contributions in the regions of
the Si atomic planes both below the Fermi energy $E_{F}$ (occupied states) and
above the insulating band gap (unoccupied states). The location of the
molecular orbitals of the F6-TCNNQ molecule both in real space as well as on
the energy axis can be made visible with this method. In addition, the plane-
averaged LSDOS gives insight into the magnetic properties of the system. While
there is no magnetization visible for the Si states, the orbitals of the
attached molecule appear to be spin-polarized. Note that this is in agreement
with Fig. 4c of Ref. [14] and Fig. 4 of Ref. [12].

Figure 2: LSDOS (top view) of the hydrogenized Si(111) slab with the F6-TCNNQ
molecule adsorbed to the surface as a constart-value contour at $E=E_{F}+$0.05
eV. The data (yellow) is shown as a contour plot. The color code of the atoms
is the same as in the bottom panel of Fig. 1. The figure was created with
VESTA 3.4.6 [15].

Complementary to the plane-averaged data, we also calculated the full LSDOS
using the LSDOSFULLMAG routine of DensityTool. We then set the energy to
$E=E_{F}+$0.05 eV and visualized the LSDOS as a contour surface of constant
spin polarization shown in Fig. 2. In agreement with the central panel of Fig.
1, the LSDOS is significantly positive for the chosen energy only in the
region around the molecule.

###  4.2 Perovskite slab model:

In this example we studied a slab model of the CH3NH3PbI3 perovskite, also
known as MAPI. Particularly, we used ($2\times 2)\times 6$ slabs of (001) MAI-
and PbI2(unpolarized)-terminated surfaces of MAPI, see Ref. [16] for slab
nomenclature. Electronic structure was calculated with the HSE06 hybrid
functional [17, 18] and spin-orbit coupling was included. For more calculation
details see the supporting information in Ref. [16]. The atomic structure for
both slabs is shown on top of Fig. 3.

Figure 3: LDOS of MAI- (left) and PbI2(unpolarized)-terminated (right) surface
slabs. The atomic structure of each slab is added on top of each LDOS figure
as a visual guide. The figures were created with Matplotlib v2.2.5, VESTA
v3.4.3 [15] and Inkscape 0.92.3.

Using DensityTool, we calculated the LDOS of both systems in the non-collinear
setting. For this case, the partial charge density written by VASP is the
starting point. Only its average in the plane parallel to the slab surface is
needed and it is calculated with the PCA routine. Subsequently, the plane-
averaged LDOS is calculated around the band gap with the LDOS routine. We
present the results in the left and right panel of Fig. 3 for MAI- and
PbI2-terminated slabs, respectively. From both plane-averaged LDOS profiles,
it is evident that the electronic structure is highly correlated with the
structure of alternated MAI and PbI2 layers of this material. Particularly,
the inner slab (bulk-like) states near the band gap (valence state maximum and
conduction states) are composed mainly of PbI2 layers. The PbI2-terminated
slab exhibits surface states located at the surface PbI2 layers, which
decrease the band gap of the entire system. This does not affect the inner
slab (bulk-like) band gap, which is almost the same as the inner band gap of
MAI-terminated slab. Finally, PbI2-terminated slab states are remarkably
deeper in energy ($\sim$1.3 eV), which means the magnitude of the ionization
potential of this exposed surface is larger than for the MAI-terminated one.
These conclusions are consistent with Ref. [16].

##  5 Conclusion

With DensityTool we provide a practical post-processor to VASP, which allows
for the manipulation and visualization of the partial charge density in the
form of LDOS or LSDOS, which can be also averaged in a plane spanned by two of
the three lattice vectors of the periodic unit cell. This approach is
complementary and in many cases superior to the commonly used approach of
projecting the DOS on the atomic spheres. Using the partial charge density and
DensityTool, a better spatial resolution can be achieved and also vacuum
regions can be studied. The L(S)DOS in real space can be also visualized as a
constant-value contour for chosen energy values.

DensityTool takes directly the output of VASP as input. The plane-averaged
L(S)DOS is written to a set of files (one for each energy value) and it can be
plotted with any common plotting program. Full L(S)DOS is written in the VASP
format (e.g. the one of CHGCAR) for compatibility. The data can thus be easily
visualized using e.g. VESTA [15].

We believe that DensityTool will be utilized to improve the understanding of
the local electronic structure of inhomogeneous systems, such as interfaces,
defects, surfaces, adsorbed molecules, or hybrid inorganic-organic composites.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgment

We thank Prof. Silvana Botti, Prof. Eduardo Menéndez-Proupin, and Prof.
Francisco Muñoz for support and fruitful discussions. T. Rauch acknowledges
funding from the Volkswagen Stiftung (Momentum) through the project
“dandelion”.

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# The log-Minkowski inequality of curvature entropy for non-symmetric convex
bodies

Chunna Zeng1, Xu Dong1, Yaling Wang1, Lei Ma2∗ 1.School of Mathematical
Sciences, Chongqing Normal University, Chongqing 401331, People’s Republic of
China<EMAIL_ADDRESS>1.School of Mathematical Sciences, Chongqing Normal
University, Chongqing 401331, People’s Republic of China
<EMAIL_ADDRESS>1.School of Mathematical Sciences, Chongqing Normal
University, Chongqing 401331, People’s Republic of China<EMAIL_ADDRESS>2.School of Sciences, Guangdong Preschool Normal College in Maoming, Maoming
525200, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

In an earlier paper [29] the authors introduced the notion of curvature
entropy, and proved the plane log-Minkowski inequality of curvature entropy
under the symmetry assumption. In this paper we demonstrate the plane log-
Minkowski inequality of curvature entropy for general convex bodies. The
equivalence of the uniqueness of cone-volume measure, the log-Minkowski
inequality of volume, and the log-Minkowski inequality of curvature entropy
for general convex bodies in $\mathbb{R}^{2}$ are shown.

Keywords: the log-Minkowski inequality of curvature entropy, log-Minkowski
inequality, dilation position, uniqueness for the cone-volume measures.

The first author was supported in part by the Major Special Project of NSFC
(Grant No. 12141101), the Young Top-Talent program of Chongqing(Grant No.
CQYC2021059145), NSF-CQCSTC(Grant No. cstc2020jcyj-msxmX0609), Technology
Research Foundation of Chongqing Educational committee(Grant No.
KJQN201900530, KJZD-K202200509), the Research Project of Chongqing Education
Commission CXQT21014.

The second author was supported by the Characteristic innovation projects of
universities in Guangdong province (Grant No. 2020KTSCX358).

*Corresponding author: Lei Ma

##  1\. Introduction

The Brunn-Minkowski theory of convex bodies, also known as the Minkowski mixed
volume theory, is the center of convex geometric analysis. In fact the Brunn-
Minkowski theory is the combination of the vector addition and volume. This
point is fully demonstrated in Minkowski’s paper [32]. Every development of
this theory just reflects this point: once the addition of convex bodies is
extended, the Brunn-Minkowski theory will evolve into a new stage. The theory
has experienced the classical stage, the $L_{p}$ stage, and the Orlicz stage.
Over a hundred years, the Brunn-Minkowski theory has established important
connections among many analytical branches in mathematics, such as probability
and statistic, information theory, physics, ellipsoidal partial differential
equations, algebraic geometry and so on (see [20] for more detail).

The Brunn-Minkowski inequality is a fundamental geometric inequality in the
classical Brunn-Minkowski theory. It gives the log-concavity of the Lebesgue
measure and implies the classical isoperimetric inequality in the Euclidean
space. Gardner’s excellent survey article [20] describes its generalizations,
consequences, and applications in geometry, analysis, probability, and other
subjects. During the last two decades, motivated by the study of geometry of
$L_{p}$ spaces, the $L_{p}$ Brunn-Minkowski theory has achieved enormous
success when $p>1$. The classical Brunn-Minkowski theory corresponds to the
case of $p=1$. The case of $p<1$, in particular, the singular case $p=0$,
remains a challenge.

One of the most important problems in the $L_{p}$ Brunn-Minkowski theory is to
establish the corresponding Brunn-Minkowski inequality when $p<1$. In fact,
the singular case $p=0$ is strongest and implies all cases of $p>0$. Following
the breakthrough of the work [8], the authors of [8] proved in [7] the Brunn-
Minkowski inequality for the case $p=0$ in the plane which is called the log-
Brunn-Minkowski inequality. The higher dimensional case is still open and is
considered as a major problem in convex geometry.

The logarithmic Minkowski inequality of volume. In [7], Böröczky, Lutwak, Yang
and Zhang conjectured that for origin-symmetric convex bodies $K$, $L$ in
$\mathbb{R}^{n}$ and $\lambda\in(0,1)$, then

$\displaystyle V((1-\lambda)K+_{0}\lambda L)\geq
V(K)^{1-\lambda}V(L)^{\lambda}.$ (1.1)

(1.1) is called log-Brunn-Minkowski inequality and note that it is not true
for general convex bodies. The log-Brunn-Minkowski inequality is stronger than
its classical counterpart for origin-symmetric convex bodies. Böröczky,
Lutwak, Yang and Zhang also demonstrated that (1.1) is equivalent to the
following log-Minkowski inequality in $\mathbb{R}^{n}$

$\displaystyle\int_{\mathbb{S}^{n-1}}\emph{\emph{log}}\left(\frac{h_{L}}{h_{K}}\right)dV_{K}\geq\frac{V(K)}{n}\emph{\emph{log}}\left(\frac{V(L)}{V(K)}\right),$
(1.2)

where $V_{K}$ is the cone volume measure of $K$. Furthermore, Böröczky,
Lutwak, Yang and Zhang proved (1.2) in planar case by using the uniqueness of
cone-volume measures.

So far, the log-Minkowski inequality and its extensions have generated a
considerable literature. For instance, Rotem [36] solved the log-Minkowski
inequality in the complex space. Saroglou [37] verified the conjecture of
(1.2) when $K$ and $L$ are both simultaneously unconditional with respect to
the same orthogonal basis, meaning that they are invariant under reflections
with respect to the principle coordinate hyperplanes $x_{i}=0$. Colesanti,
Livshyts and Marsiglietti [13] verified (1.2) locally for small-enough
$C^{2}$-perturbations of the Euclidean ball $B$. Xi and Leng [46] proved Dar’s
conjecture and demonstrated the equivalence between the log-Brunn-Minkowski
inequality and the log-Minkowski inequality for non-symmetric convex bodies.
Tao Xiong and Xiong [42] proved the log-Minkowski inequality for cylinders in
$\mathbb{R}^{3}.$ Ma, Zeng and Wang [29] showed that the log-Minkowski
inequality of volume is equivalent to the log-Minkowski inequality of
curvature entropy in $\mathbb{R}^{n}.$

The $L_{p}$ Minkowski problem. The $L_{p}$ Minkowski problem is one of the
central problems in contemporary convex geometric analysis. The classical
Minkowski problem states that: given a finite Borel measure $\mu$ on
$\mathbb{S}^{n-1}$, what are the necessary and sufficient conditions so that
$\mu$ is the surface area measure of a convex body $K$? Minkowski [31] solved
this question when the given measure is either discrete or has a continuous
density. Later, Aleksandrov [2, 3], Fenchel and Jessen [14] solved the problem
for general measure. It showed that if $\mu$ is not concentrated on any closed
hemispherical surface, then $\mu$ is the surface area measure of $K$ when and
only when its centroid is at the origin. The $L_{p}$ Minkowski problem is an
extension of the classical Minkowski problem and has achieved great
developments. The $L_{p}$ Minkowski problem in the plane is solved by Stancu
[40, 41], Umanskiy [44], Chen [11], and Jiang [25]. The solution of $L_{p}$
Minkowski problem is homothetic solutions of Gauss curvature flow, see [5, 6,
12, 18, 43]. When $\mu$ is Lebesgue measure on unit circle $\mathbb{S}^{1}$,
then the solution of $L_{p}$ Minkowski problem in $\mathbb{R}^{2}$ is
homothetic solutions of misdirected curve flow, see Andrews [6]. Obviously,
the $L_{1}$ Minkouski problem is the classical Minkowski problem, while the
$L_{0}$ Minkowski problem is the logarithmic Minkowski problem.

The conjectured uniqueness of log-Minkowski problem. The log-Minkowski problem
is the most important case because the cone-volume measure is the only $SL(n)$
invariant measure among all the $L_{p}$ surface area measure.

[_Lutwak_] If $K$ and $L$ are symmetric smooth strictly convex sets with
$V_{K}(\omega)=V_{L}(\omega)$, where $\omega\in\mathbb{S}^{n-1}$ is a Borel
set, then $K=L$.

Firey [16] proved that if the cone-volume measure of a origin-symmetric convex
body is a positive constant multiplied by spherical Lebesgue measure in
$\mathbb{R}^{n}$, then the body must be an Euclidean ball. In [8], Böröczky,
Lutwak, Yang and Zhang showed that if $K$, $L$ are origin-symmetric and have
the same cone-volume measures in $\mathbb{R}^{2}$, then they are either
parallelograms with parallel sides or $K=L$. The special case of smooth
origin-symmetric planar convex bodies with positive curvature was proved by
Gage [17]. A nature question is whether the uniqueness of cone-volume measure
holds without symmetrical condition. In [46], Xi and Leng gave the definition
of dilation position for the first time to prove the log-Brunn-Minkowski
inequality for two convex bodies $K,L\in\mathcal{K}_{0}^{2}$ and solved the
planar Dar’s conjecture. Zhu [50] and Stancu [40] solved the case of discrete
measure. However, the uniqueness condition is still remain open in higher
dimensional case.

The log-Minkowski inequality of curvature entropy for symmetric convex bodies.
There are many entropy inequalities which have deep relationships with the
Brunn-Minkowski inequality, such as Cramer-Rao inequalities, Fisher
information inequality, moment-entropy inequality, entropy power inequality,
Stams inequality and so on, see [26, 27, 30]. For example, the entropy power
inequality states that the effective variance (entropy power) of the sum of
two independent random variables is greater than the sum of their effective
variances. While the Brunn-Minkowski inequality states that the effective
radius of the set sum of two sets is greater than the sum of their effective
radii. Both these inequalities are recast in a form that enhances their
similarity.

In [29], the authors introduced the notion of curvature entropy in
$\mathbb{R}^{n}$. Assume that $K,~{}L\in\mathcal{K}^{n}_{0}$, then the
curvature entropy $E\left(K,L\right)$ is defined as

$E(K,L)=-\int_{\mathbb{S}^{n-1}}\log\frac{H_{n-1}\left(L\right)}{H_{n-1}\left(K\right)}dV_{K},$
(1.3)

where $H_{n-1}\left(\cdot\right)$ denotes the Gauss curvature of the boundary
of a convex body. They obtained the plane log-Minkowski inequality of
curvature entropy in $\mathbb{R}^{2}$ when one convex body is symmetric. And
they also demonstrated that under symmetry condition, the uniqueness of cone-
volume measure, the log-Minkowski inequality of volume and the log-Minkowski
inequality of curvature entropy are equivalent in $\mathbb{R}^{n}$. They
conjectured the following problem

Problem 1. Suppose that $K,L\in\mathcal{K}^{n}_{0}$ are strictly convex bodies
of $C^{2}$ boundaries with $K$ symmetric in $\mathbb{R}^{n}$, then

$E(K,L)\leq\frac{n-1}{n}V(K)\log\frac{V(L)}{V(K)},$ (1.4)

with equality holds when and only when $K$ and $L$ are homothetic.

By using techniques offered by [46] and [47], the authors obtain the following
plane log-Minkowski inequality of curvature entropy for general convex bodies
without the symmetry assumption.

######  Theorem 1.

Let $K,L\in\mathcal{K}_{0}^{2}$ be two smooth, planar strictly convex bodies.
If $K$ and $L$ are at a dilation position, then

$E(K,L)\leq\frac{V(K)}{2}\log\frac{V(L)}{V(K)},$ (1.5)

the equality holds when and only when $K$ and $L$ are homothetic.

As an application of above theorem, we obtain the following uniqueness
condition for planar cone-volume measure of non-symmetric convex bodies.

######  Theorem 2.

Let $K,L\in\mathcal{K}_{0}^{2}$ be two smooth, planar strictly convex bodies.
If $K$ and $L$ are at a dilation position and have the same cone-volume
measure, then $K=L.$

One of main aims of this paper is to show that for two general convex bodies
in $\mathbb{R}^{2}$, the log-Minkowski inequality of volume and the log-
Minkowski inequality of curvature entropy (1.5) are equivalent. Furthermore,
the equivalence of the uniqueness of cone-volume measure, the log-Minkowski
inequality of volume and the log-Minkowski inequality of curvature entropy are
established in $\mathbb{R}^{2}$.

######  Theorem 3.

Let $K,L\in\mathcal{K}_{0}^{2}$ be two smooth, planar strictly convex bodies.
If $K$ and $L$ are at a dilation position, then the uniqueness for cone-volume
measures, the log-Minkowski inequality of volume and the log-Minkowski
inequality of curvature entropy are equivalent.

##  2\.  Preliminaries

In this section, we list some notation and basic facts about convex bodies.
Good general references for the theory of convex bodies see, e.g., [19, 23,
39].

Write $x\cdot y$ for the standard inner product of $x,y\in\mathbb{R}^{n}$. A
_convex body_ in $\mathbb{R}^{n}$ is a compact convex subset with non-empty
interior in $\mathbb{R}^{n}$. Denote by $\mathcal{K}^{n}$ and
$\mathcal{K}^{n}_{0}$ the set of convex bodies and convex bodies containing
the origin in their interior in $\mathbb{R}^{n}$. A convex body $K$ is
uniquely determined by its support function
$h_{K}:\mathbb{R}^{n}\rightarrow\mathbb{R}$,

$h_{K}(x)=\max\left\\{x\cdot y:y\in K\right\\}.$

The support function is positively homogeneous of degree one and subadditive.

Assume that $K$ is a strictly convex body of $C^{2}$ boundary in
$\mathbb{R}^{n}$. Let $u:\partial K\rightarrow\mathbb{S}^{n-1}$ be the Gauss
map. Then the surface area element $dS(x)$ of $\partial K$ and the surface
area element of $\mathbb{S}^{n-1}$ at $u$ are related by

$dS(x)=\frac{1}{H_{n-1}}du,$ (2.1)

where $H_{n-1}$ is the Gauss curvature of $\partial K$ at $x$. By the
following expression of the reciprocal Gauss curvature ([48])

$\frac{1}{H_{n-1}}=\det(h_{ij}+h_{K}\delta_{ij}),$

where $h_{ij}$ is the covariant derivative of $h$ with respect to an
orthonormal frame on $\mathbb{S}^{n-1}$ and $\delta_{ij}$ is the Kronecker
delta, (2.1) can be rewritten as

$dS(x)=\det(h_{ij}+h_{K}\delta_{ij})du.$

Thus

$V\left(K\right)=\frac{1}{n}\int_{\mathbb{S}^{n-1}}h_{K}\det(h_{ij}+h_{K}\delta_{ij})du.$

The relative curvature radius of $K$ with respect to $L$ in $\mathbb{R}^{n}$
is the ratio of Gauss curvature of $K$ and $L$, which is defined by

$\rho_{K,L}=\frac{H_{n-1}\left(L\right)}{H_{n-1}\left(K\right)}=\frac{det\left(h_{ij}\left(K\right)+h_{K}\delta_{ij}\right)}{det\left(h_{ij}\left(L\right)+h_{L}\delta_{ij}\right)}.$
(2.2)

Specially, in the plane case,

$\rho_{K,L}=\frac{\kappa_{L}}{\kappa_{K}}=\frac{h_{K}+h_{K}^{\prime\prime}}{h_{L}+h_{L}^{\prime\prime}},$
(2.3)

where $\kappa$ denotes the curvature of closed convex curve. If $K$ and $L$
are homothetic, then $\rho_{K,L}$ is a constant.

For $K,L\in\mathcal{K}^{n}$, the Minkowski addition is defined by
$K+L=\\{x+y:x\in K,y\in L\\},$ and the multiple of $K$ by the scalar
$\lambda>0$ is $\lambda K=\\{\lambda x:x\in K\\}.$ The Minkowski-Steiner
formula states that

$\displaystyle V(K+\lambda L)=\sum\limits_{i=0}^{n}\left(\begin{aligned} &n\\\
&i\end{aligned}\right)V_{i}(K,L)\lambda^{i},$ (2.4)

where $\lambda>0.$ $V(\cdot)$ is the $n$-dimensional volume and the
coefficients

$V_{i}(K,L)=V(\underbrace{K,\cdots,K}_{n-i}\underbrace{L,\cdots,L}_{i})$

are called _mixed volumes_ of $K$ and $L,$ which are nonnegative, symmetric in
the indices, multilinear, translation invariant and homogeneous.

Actually, (2.4) can be rewritten as the following form, which gives the
relative Steiner polynomial of $K$ with respect to $L$, that is

$V\left(K+tL\right)=V(K)+2V(K,L)t+V(L)t^{2}.$ (2.5)

From the Minkowski inequality

$V^{2}(K,L)-V(K)V(L)\geq 0,$

it follows that the expression $V(K+tL)=0$ has two negative real roots. Denote
by $t_{1}$ and $t_{2}$ ($t_{1}>t_{2}$) the two roots of the relative Steiner
polynomial of $K$ with respect to $L$

$t_{1}=-\frac{V(K,L)}{V(L)}+\frac{\sqrt{V(K,L)^{2}-V(K)V(L)}}{V(L)},$ (2.6)
$t_{2}=-\frac{V(K,L)}{V(L)}-\frac{\sqrt{V(K,L)^{2}-V(K)V(L)}}{V(L)}.$ (2.7)

The following Lemma introduced nonsymmetric extension of _Green-Osher
inequality_.

######  Lemma 1.

([47]) Let $K$ and $L$ be two smooth, planar strictly convex bodies and
$\rho(\theta)$ the relative curvature radius of $K$ with respect to $L.$ If
$K$ and $L$ are at a dilation position and $F(x)$ is a strictly convex
function on $(0,+\infty),$ then

$\frac{1}{V(L)}\int^{2\pi}_{0}F(\rho(\theta))h_{L}(\theta)(h_{L}(\theta)+h_{L}^{\prime\prime}(\theta))d\theta\geq
F(-t_{1})+F(-t_{2}),$ (2.8)

where $t_{1},$ $t_{2}$ are the two roots of the relative Steiner polynomial of
$K$ with respect to $L,$ and the equality holds when and only when $K$ and $L$
are homothetic.

The cone-volume measure $V_{K}$ of $K\in\mathcal{K}^{n}_{0}$ is a Borel
measure on the unit sphere $\mathbb{S}^{n-1}$ defined for a Borel set
$\omega\subseteq\mathbb{S}^{n-1}$, by

$\displaystyle V_{K}\left(\omega\right)=\frac{1}{n}\int_{x\in
v_{K}^{-1}\left(\omega\right)}{x\cdot
v_{k}\left(x\right)d\mathcal{H}^{n-1}}\left(x\right),$

where $v_{K}:\partial K\rightarrow\mathbb{S}^{n-1}$ is the generalized Gauss
map defined on $\partial K$, $\mathcal{H}^{n-1}$ is $(n-1)$-dimensional
Hausdorff measure. There are formulas

$dV_{K}=\frac{1}{n}h_{K}dS_{K},$

and

$V(K)=\frac{1}{n}\int_{\mathbb{S}^{n-1}}h_{K}(u)dS_{K}(u).$

The mixed cone-volume measure $V_{K,L}$ of $K,L\in\mathcal{K}_{0}^{n}$ is
introduced by Hu and Xiong [24]

$V_{K,L}\left(\omega\right)=\frac{1}{n}\int_{\omega}h_{L}(u)dS_{K}.$

The total mass of $V_{K,L}$ is exactly the 1st mixed volume $V_{1}(K,L)$. In
the Euclidean plane, it is clear that $V_{1}(K,L)=V_{1}(L,K)$, and we write
$V(K,L)$ instead of $V_{1}(K,L)$.

Let $K$, $L\in\mathcal{K}^{n}$, $K$ and $L$ are at a _dilation position_ if
the origin $o\in K\cap L$ and

$r(K,L)L\subset K\subset R(K,L)L.$

where $r(K,L)$ and $R(K,L)$ are the inradius and outradius of $K$ with respect
to $L$, i.e.

$r\left(K,L\right)=\max\\{t>0:x+tL\subset K~{}and~{}x\in\mathbb{R}^{n}\\},$
$R\left(K,L\right)=\min\\{t>0:x+tL\supset K~{}and~{}x\in\mathbb{R}^{n}\\}.$

The concept of dilation position is based on solving the problem provided by
G. Zhang in 2013, that is: Let $K,L\in\mathcal{K}^{2}$, if there is a
“suitable” position of the origin so that $K$ and a “suitable” translation of
$L$ satisfy

$V((1-\lambda)K+_{0}\lambda L)\geq V(K)^{1-\lambda}V(L),$

where $K$ and $L$ are origin-symmetric convex bodies in the plane.

The following is the general log-Minkowski inequality for planar convex
bodies.

######  Lemma 2.

([46]) Let $K,L\in\mathcal{K}^{2}$ with $o\in K\cap L.$ If $K$ and $L$ are at
a dilation position, then

$\int_{\mathbb{S}^{1}}\log\frac{h_{L}}{h_{K}}dV_{K}\geq\frac{V(K)}{2}\log\frac{V(L)}{V(K)}.$
(2.9)

The equality holds when and only when $K$ and $L$ are dilates or $K$ and $L$
are parallelograms with parallel sides.

##  3\.  the log-Minkowski inequality of curvature entropy with nonsymmetry in
$\mathbb{R}^{2}$

_Proof of Theorem 1._ Let $F(x)=-\log x$. By Lemma 1, (2.6) and (2.7), it
follows that

$\displaystyle\frac{1}{V(L)}\int^{2\pi}_{0}(-\log\frac{h_{K}+h_{K}^{\prime\prime}}{h_{L}+h_{L}^{\prime\prime}})h_{L}(\theta)(h_{L}(\theta)+h_{L}^{\prime\prime}(\theta))d\theta$
$\displaystyle\geq-\log(-t_{1})-\log(-t_{2})$
$\displaystyle=-\log(t_{1}t_{2})$ $\displaystyle=-\log\frac{V(K)}{V(L)}.$

That is

$-\frac{2}{V(L)}\int_{\mathbb{S}^{1}}\log\frac{h_{K}+h_{K}^{\prime\prime}}{h_{L}+h_{L}^{\prime\prime}}dV_{L}\geq-\log\frac{V(K)}{V(L)}.$

Thus

$\int_{\mathbb{S}^{1}}\log(\frac{h_{K}+h_{K}^{\prime\prime}}{h_{L}+h_{L}^{\prime\prime}})dV_{L}\leq\frac{V(L)}{2}\log\frac{V(K)}{V(L)}.$

By (1.3), (2.2) and (2.3) for planar case, we have

$E(L,K)\leq\frac{V(L)}{2}\log\frac{V(K)}{V(L)}.$ (3.1)

Note that $K$ and $L$ are at a dilation position, so

$E(K,L)\leq\frac{V(K)}{2}\log\frac{V(L)}{V(K)}.$ (3.2)

From the equality condition of (2.8), it concludes that the equality holds in
(1.5) when and only when $K$ and $L$ are homothetic. ∎

Especially, assume that $L=B$ is an unit Euclidean ball in $\mathbb{R}^{2}$,
then

######  Corollary 1.

([22]) Let $K\in\mathcal{K}_{0}^{2}$ be smooth, planar strictly convex bodies,
then

$\int_{\partial K}\kappa\log\kappa ds+\pi\log\frac{V(K)}{\pi}\geq 0,$ (3.3)

the equality holds when and only when $K$ is a ball.

###### Proof.

According to the definition of $E(L,K)$, it follows that

$\begin{split}E(L,K)&=-\int_{\mathbb{S}^{n-1}}\log\frac{H_{n-1}(K)}{H_{n-1}(L)}dV_{L}\\\
&=-\frac{1}{n}\int_{\mathbb{S}^{n-1}}\log\frac{H_{n-1}(K)}{H_{n-1}(L)}h_{L}\det\big{(}h_{ij}(L)+h_{L}\delta_{ij}\big{)}du\\\
&=-\frac{1}{n}\int_{\mathbb{S}^{n-1}}\log\frac{H_{n-1}(K)}{H_{n-1}(L)}h_{L}\frac{\det\big{(}h_{ij}(L)+h_{L}\delta_{ij}\big{)}}{\det\big{(}h_{ij}(K)+h_{K}\delta_{ij}\big{)}}\det\big{(}h_{ij}(K)+h_{K}\delta_{ij}\big{)}du\\\
&=-\int_{\mathbb{S}^{n-1}}\frac{H_{n-1}(K)}{H_{n-1}(L)}\log\frac{H_{n-1}(K)}{H_{n-1}(L)}dV_{K,L}.\end{split}$

Taking $n=2$ and $L=B$, it yields

$\begin{split}E(B,K)&=-\int_{\mathbb{S}^{1}}{\kappa}\log\kappa dV_{K,B}\\\
&=-\frac{1}{2}\int_{\partial K}{\kappa}\log\kappa ds\\\
&\leq\frac{\pi}{2}\log\frac{V(K)}{\pi}.\end{split}$ (3.4)

It follows that

$\int_{\partial K}\kappa\log\kappa ds+\pi\log\frac{V(K)}{\pi}\geq 0.$

∎

######  Corollary 2.

([22]) Let $K\in\mathcal{K}_{0}^{2}$ be smooth, planar strictly convex bodies.
Then

$\int_{\partial K}\kappa\log\left(\kappa\sqrt{\frac{V(K)}{\pi}}\right)ds\geq
0,$ (3.5)

the equality holds when and only when $K$ is a ball.

###### Proof.

Taking $ds=(h+h^{\prime\prime})d\theta$, $\kappa=\frac{1}{h+h^{\prime\prime}}$
and $\int_{\partial K}d\theta=2\pi$ in (3.3), we have

$\displaystyle\int_{\partial K}\kappa\log\kappa ds+\pi\log\frac{V(K)}{\pi}$
$\displaystyle=\int_{\partial K}\kappa\log\kappa
ds+\frac{1}{2}\log\frac{V(K)}{\pi}\int_{\partial K}d\theta$
$\displaystyle=\int_{\partial K}\kappa\log\kappa ds+\frac{1}{2}\int_{\partial
K}\frac{1}{h+h^{\prime\prime}}(h+h^{\prime\prime})\log\frac{V(K)}{\pi}d\theta$
$\displaystyle=\int_{\partial K}\kappa\log\kappa ds+\frac{1}{2}\int_{\partial
K}\kappa\log\frac{V(K)}{\pi}ds$ $\displaystyle=\int_{\partial
K}\kappa\log\left(\kappa\sqrt{\frac{V(K)}{\pi}}\right)ds$ $\displaystyle\geq
0.$

∎

##  4\. The uniqueness for the planar Cone-volume Measures with nonsymmetry

_Proof of Theorem 2._ Assuming $V_{K}=V_{L},$ it is obvious that $V(K)=V(L).$
From (3.1) we have

$E(L,K)\leq\frac{V(L)}{2}\log{\frac{V(K)}{V(L)}}=0,$

that is

$\int_{\mathbb{S}^{1}}\log(h_{K}+h^{\prime\prime}_{K})dV_{L}\leq\int_{\mathbb{S}^{1}}\log(h_{L}+h^{\prime\prime}_{L})dV_{L}.$
(4.1)

Similarly, by (3.2)

$E(K,L)\leq\frac{V(K)}{2}\log{\frac{V(L)}{V(K)}}=0,$

which is equivalent to

$\int_{\mathbb{S}^{1}}\log(h_{L}+h^{\prime\prime}_{L})dV_{K}\leq\int_{\mathbb{S}^{1}}\log(h_{K}+h^{\prime\prime}_{K})dV_{K}.$
(4.2)

By (4.1), (4.2) and $V_{K}=V_{L},$ then

$\displaystyle\int_{\mathbb{S}^{1}}\log(h_{L}+h^{\prime\prime}_{L})dV_{K}$
$\displaystyle\leq\int_{\mathbb{S}^{1}}\log(h_{K}+h^{\prime\prime}_{K})dV_{K}$
$\displaystyle=\int_{\mathbb{S}^{1}}\log(h_{K}+h^{\prime\prime}_{K})dV_{L}$
$\displaystyle\leq\int_{\mathbb{S}^{1}}\log(h_{L}+h^{\prime\prime}_{L})dV_{L}$
$\displaystyle=\int_{\mathbb{S}^{1}}\log(h_{L}+h^{\prime\prime}_{L})dV_{K}.$

So

$\int_{\mathbb{S}^{1}}\log(h_{L}+h^{\prime\prime}_{L})dV_{K}=\int_{\mathbb{S}^{1}}\log(h_{K}+h^{\prime\prime}_{K})dV_{K}.$

It yields

$E(K,L)=\frac{V(K)}{2}\log\frac{V(L)}{V(K)}=0.$

By Theorem 1, it follows that $K$ and $L$ are homothetic. Because of
$V(K)=V(L),$ we conclude that $K=L.$ ∎

##  5\. Equivalence between uniqueness for cone-volume measures, log-Minkowski
inequality of volume and log-Minkowski inequality of curvature entropy in
$\mathbb{R}^{2}$

_Proof of Theorem 3._ By Lemma 2, if $K,L\in\mathcal{K}^{2}_{0}$ are at a
dilation position, it concludes that the uniqueness of the cone volume measure
infers the log-Minkowski inequality in $\mathbb{R}^{2}$. In order to prove
that the uniqueness for cone-volume measures, the log-Minkowski inequality of
volume and the log-Minkowski inequality of curvature entropy are equivalent in
$\mathbb{R}^{2}$, we only need to prove that the log-Minkowski inequality of
volume can deduce the log-Minkowski inequality of curvature entropy.

By the definition of $E(K,L)$, we have

$\displaystyle\int_{\mathbb{S}^{1}}\log\frac{h_{L}}{h_{K}}dV_{K}+E(K,L)$
$\displaystyle=\int_{\mathbb{S}^{1}}\log\frac{h_{L}(h_{L}+h^{\prime\prime}_{L})}{h_{K}(h_{K}+h^{\prime\prime}_{K})}dV_{K}$
(5.1) $\displaystyle\leq
V(K)\log\left(\frac{\int_{\mathbb{S}^{1}}\frac{h_{L}(h_{L}+h^{\prime\prime}_{L})}{h_{K}(h_{K}+h^{\prime\prime}_{K})}dV_{K}}{V(K)}\right)$
$\displaystyle=V(K)\log\frac{V(L)}{V(K)},$

where the second step in (5.1) is due to Jensen’s inequality. It follows that

$\int_{\mathbb{S}^{1}}\log\left(\frac{h_{L}}{h_{K}}\right)dV_{K}\leq
V(K)\log\frac{V(L)}{V(K)}-E(K,L).$

By the log-Minkowski inequality (2.9), we obtain

$\frac{V(K)}{2}\log\frac{V(L)}{V(K)}\leq\int_{\mathbb{S}^{1}}\log\left(\frac{h_{L}}{h_{K}}\right)dV_{K}\leq
V(K)\log\frac{V(L)}{V(K)}-E(K,L).$ (5.2)

Then

$E(K,L)\leq\frac{V(K)}{2}\log\frac{V(L)}{V(K)}.$

Next, we consider the equality condition of the log-Minkowski inequality for
curvature entropy. Suppose that

$E(K,L)=\frac{V(K)}{2}\log\frac{V(L)}{V(K)}.$

From (5.2) it leads to

$\frac{V(K)}{2}\log\frac{V(L)}{V(K)}\leq\int_{\mathbb{S}^{1}}\log\left(\frac{h_{L}}{h_{K}}\right)dV_{K}\leq\frac{V(K)}{2}\log\frac{V(L)}{V(K)}.$

i.e.

$\int_{\mathbb{S}^{1}}\log\left(\frac{h_{L}}{h_{K}}\right)dV_{K}=\frac{V(K)}{2}\log\frac{V(L)}{V(K)},$

then $K$ and $L$ are dilates. In addition, if $K$ and $L$ are dilates,
$E(K,L)=\frac{V(K)}{2}\log\frac{V(L)}{V(K)}$ obviously holds. ∎

##  6\.  The entropy inequality in $\mathbb{R}^{n}$ and some remarks

In this section, an entropy inequality under a more general condition in
$\mathbb{R}^{n}$ is established. The conditions of dilation position and
symmetry are removed.

_Proof of Theorem LABEL:xx._ Note that

$\displaystyle\int_{\mathbb{S}^{n-1}}\frac{H_{n-1}(L)}{H_{n-1}(K)}\log{\frac{H_{n-1}(L)}{H_{n-1}(K)}}dV_{L,K}=\int_{\mathbb{S}^{n-1}}\log{\frac{H_{n-1}(L)}{H_{n-1}(K)}}dV_{K}.$
(6.1)

By Lebesgue’s dominated convergence theorem, as $p\rightarrow\infty,$

$\displaystyle\int_{\mathbb{S}^{n-1}}\left(\frac{H_{n-1}(L)}{H_{n-1}(K)}\right)^{\frac{p}{p+n}}dV_{L,K}\rightarrow
V(K),$

and

$\displaystyle\int_{\mathbb{S}^{n-1}}\left(\frac{H_{n-1}(L)}{H_{n-1}(K)}\right)^{\frac{p}{p+n}}\log{\frac{H_{n-1}(L)}{H_{n-1}(K)}}dV_{L,K}\rightarrow\int_{\mathbb{S}^{n-1}}\left(\frac{H_{n-1}(L)}{H_{n-1}(K)}\right)\log{\frac{H_{n-1}(L)}{H_{n-1}(K)}}dV_{L,K}.$

Consider the function $f_{K,L}:[1,\infty]\rightarrow\mathbb{R}$ defined by

$\displaystyle
f_{K,L}(p)=\frac{1}{V(K)}\int_{\mathbb{S}^{n-1}}\left(\frac{H_{n-1}(L)}{H_{n-1}(K)}\right)^{\frac{p}{p+n}}dV_{L,K},$

and calculate, using L’H$\hat{o}$pital’s rule,

$\displaystyle\lim_{p\rightarrow\infty}\log(f_{K,L}(p))^{p+n}$
$\displaystyle=\lim_{p\rightarrow\infty}\frac{\log{f_{K,L}(p)}}{\frac{1}{p+n}}$
$\displaystyle=\lim_{p\rightarrow\infty}\frac{f^{{}^{\prime}}_{K,L}(p)}{-\frac{f_{K,L}(p)}{(p+n)^{2}}}$
$\displaystyle=\lim_{p\rightarrow\infty}{\frac{\frac{n}{(p+n)^{2}}\int_{\mathbb{S}^{n-1}}(\frac{H_{n-1}(L)}{H_{n-1}(K)})^{\frac{p}{p+n}}\log{\frac{H_{n-1}(L)}{H_{n-1}(K)}}dV_{L,K}}{\frac{-f_{K,L}(p)V(K)}{(p+n)^{2}}}}$
$\displaystyle=-\frac{n}{V(K)}\int_{\mathbb{S}^{n-1}}\frac{H_{n-1}(L)}{H_{n-1}(K)}\log{\frac{H_{n-1}(L)}{H_{n-1}(K)}}dV_{L,K}.$

Consequently, we have

$\displaystyle\exp\left[-\frac{n}{V(K)}\int_{\mathbb{S}^{n-1}}\frac{H_{n-1}(L)}{H_{n-1}(K)}\log{\frac{H_{n-1}(L)}{H_{n-1}(K)}}dV_{L,K}\right]$
(6.2)
$\displaystyle=\lim_{p\rightarrow\infty}\left[\frac{1}{V(K)}\int_{\mathbb{S}^{n-1}}\left(\frac{H_{n-1}(L)}{H_{n-1}(K)}\right)^{\frac{p}{p+n}}dV_{L,K}\right]^{p+n}.$

Via H$\ddot{o}$lder’s inequality

$\displaystyle\left[\int_{\mathbb{S}^{n-1}}\left(\frac{H_{n-1}(L)}{H_{n-1}(K)}\right)^{\frac{p}{p+n}}dV_{L,K}\right]^{\frac{p+n}{p}}\cdot\left[\int_{\mathbb{S}^{n-1}}dV_{L,K}\right]^{-\frac{n}{p}}\leq\int_{\mathbb{S}^{n-1}}\frac{H_{n-1}(L)}{H_{n-1}(K)}dV_{L,K}=V(K),$

which is equivalent to

$\displaystyle\left[\frac{1}{V(K)}\int_{\mathbb{S}^{n-1}}\left(\frac{H_{n-1}(L)}{H_{n-1}(K)}\right)^{\frac{p}{p+n}}dV_{L,K}\right]^{\frac{p+n}{p}}\leq\frac{V(K){V_{1}(L,K)}^{\frac{n}{p}}}{{V(K)}^{\frac{p+n}{p}}}=\left[\frac{V_{1}(L,K)}{V(K)}\right]^{\frac{n}{p}}.$

It implies that

$\displaystyle\lim_{p\rightarrow\infty}\left[\frac{1}{V(K)}\int_{\mathbb{S}^{n-1}}\left(\frac{H_{n-1}(L)}{H_{n-1}(K)}\right)^{\frac{p}{p+n}}dV_{L,K}\right]^{p+n}\leq\left[\frac{V_{1}(L,K)}{V(K)}\right]^{n}.$

By (6.2), we have

$\displaystyle\exp\left[-\frac{n}{V(K)}\int_{\mathbb{S}^{n-1}}\frac{H_{n-1}(L)}{H_{n-1}(K)}\log{\frac{H_{n-1}(L)}{H_{n-1}(K)}}dV_{L,K}\right]\leq\left[\frac{V_{1}(L,K)}{V(K)}\right]^{n},$

it yields

$-\frac{n}{V(K)}\int_{\mathbb{S}^{n-1}}\frac{H_{n-1}(L)}{H_{n-1}(K)}\log{\frac{H_{n-1}(L)}{H_{n-1}(K)}}dV_{L,K}\leq
n\log\frac{V_{1}(L,K)}{V(K)},$

that is

$\displaystyle-\int_{\mathbb{S}^{n-1}}\frac{H_{n-1}(L)}{H_{n-1}(K)}\log{\frac{H_{n-1}(L)}{H_{n-1}(K)}}dV_{L,K}\leq
V(K)\log\left(\frac{V_{1}(L,K)}{V(K)}\right).$

By (6.1)

$\displaystyle-\int_{\mathbb{S}^{n-1}}\log{\frac{H_{n-1}(L)}{H_{n-1}(K)}}dV_{K}\leq
V(K)\log\left(\frac{V_{1}(L,K)}{V(K)}\right),$

thus

$\displaystyle E(K,L)\leq V(K)\log\frac{V_{1}(L,K)}{V(K)}.$

∎

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