An Optimal Estimate for the Anisotropic Logarithmic Potential

Abstract

This paper introduces the new annulus body to establish the optimal lower bound for the anisotropic logarithmic potential as the complement to the theory of its upper bound estimate which has already been investigated. The connections with convex geometry analysis and some metric properties are also established. For the application, a polynomial dual log-mixed volume difference law is deduced from the optimal estimate.

1. Backgrounds

The Riesz potential $I_{\alpha }$($\alpha >0$) operator is defined by

$$I_{\alpha }f\left(x\right)={\int }_{{\mathbb{R}}^n}\frac{f\left(y\right)}{{|x-y|}^{\alpha }}dy,$$

where f is a measurable function. It has been widely developed in harmonic analysis including function spaces, mathematical physics and partial differential equations (see [1,2,3,4]).

For the endpoint case $\alpha =0$, it is trivial to study the limitation

$$\underset{\alpha \to 0}{lim}{|x-y|}^{-\alpha }=1asx\ne y.$$

Instead, the convolution kernel is usually changed in such a derivative way

$${\left.\frac{\partial }{\partial \alpha }{|x-y|}^{-\alpha }\right|}_{\alpha =0}={\left.\frac{{log|x-y|}^{-1}}{{|x-y|}^{\alpha }}\right|}_{\alpha =0}=log{|x-y|}^{-1}asx\ne y.$$

This logarithmic kernel produces a corresponding logarithmic potential operator, which represents a the better complement for the endpoint case of Riesz potential operator by virtue of effective properties and applications. For example, ${\left|x\right|}^{2-n}(n\ge 3)$ is harmonic on ${\mathbb{R}}^n\backslash o$, while for teh lower dimension $n=2$, $log\left|x\right|$ is studied since it is harmonic on ${\mathbb{R}}^n\backslash o$ (see [5,6]).

Recently, both Riesz potential and logarithmic potential have been studied in an anisotropic way, which is closely related with convex geometry analysis and mathematical physics (see [7,8,9,10,11]). Here we first recall some basic concepts and results in convex geometry.

If the intersection of each line through the origin with a set $K⫋{\mathbb{R}}^n$ is a compact line segment, K is called star-shaped with respect to the origin. Let

$${\rho }_K\left(x\right)=\max \{\lambda \ge 0:\lambda x\in K\}\mathrm{for}x\in {\mathbb{R}}^n\backslash o,$$

where o is the origin, be the radial function of the star-shaped set K. K is called a star body with respect to the origin, if ${\rho }_K$ is positive and continuous. We assume that K is a star body with respect to the origin and E is a bounded measurable set in this paper. Note that the radial function ${\rho }_K$ is positively homogeneous with degree $-1$, i.e.,

$${\rho }_A\left(sx\right)=s^{-1}{\rho }_A\left(x\right)\mathrm{for}\mathrm{all}s>0.$$

Let $V\left(E\right)$ and $E^c$ denote, respectively, the n-dimensional volume of E and the complement of E. We assume $V\left(E\right)\ne 0$ in this paper, since when $V\left(E\right)=0$, some trivial result follows directly. Let $dS(·)$ denote the natural spherical measure on the boundary ${\mathbb{S}}^{n-1}$ of the unit ball $B_2^n$ centered at the origin. Then

$$V\left(K\right)=\frac{1}{n}{\int }_{{\mathbb{S}}^{n-1}}{\rho }_K^n\left(u\right)dS\left(u\right).$$

Let ${\parallel ·\parallel }_K$ denote by the Minkowski functional of K:

$${\parallel x\parallel }_K=inf\{s>0:x\in sK\}\mathrm{for}\mathrm{all}x\in {\mathbb{R}}^n$$

(1)

where

$$sK=\{sy:y\in K\}.$$

Note that ${\rho }_K^{-1}\left(x\right)={\parallel x\parallel }_K$ and ${\parallel ·\parallel }_{B_2^n}=|·|$, where $|·|$ denotes the Euclidean norm. We refer to [12,13] for more information on convex geometry.

Let $y\in {\mathbb{R}}^n$, $a>1$ and denote by

$$R_a^K\left(y\right)=\{x\in {\mathbb{R}}^n:\frac{1}{a}\le {\parallel x-y\parallel }_K\le a\}$$

the K-annulus body centered at y with outer radius a and inner radius $\frac{1}{a}$. Then, by the definition of the Minkowski functional, it follows that

$$V\left(R_a^K\left(y\right)\right)=\left(a^n-{\left(\frac{1}{a}\right)}^n\right)V\left(K\right).$$

Several anisotropic Riesz potentials are introduced and their optimal extreme values estimates are systematically studied in [10]. We omit the details here for the brevity of this paper. Let

$$P_{log,m}(K,E;y)={\int }_E{\left(log\frac{1}{{\parallel x-y\parallel }_K}\right)}^{m}dx$$

be the anisotropic m-log-potential of measurable set E at $y\in {\mathbb{R}}^n$ with respect to K, and

$$V_{log,m}(K,E)=\underset{y\in {\mathbb{R}}^n}{sup}{P}_{log,m}(K,E;y)$$

be the mixed volume of K and E. We refer to [11] for these definitions and [14,15] for their relations with engineering and mathematical physics.

Note that $V_{log,m}(K,E)$ is obviously an extreme value of the anisotropic m-log-potential. It is also closely related to convex geometry analysis. In [11], when m is an odd number, the optimal estimate for $V_{log,m}(K,E)$ is established as follows:

$$\begin{array}{cc}\hfill V_{log,m}(K,E)& \le \left\{\begin{array}{cc}\frac{V\left(E\right)}{n^m}\sum _{i=0}^m\frac{m!}{(m-i)!}{\left(log\frac{V\left(K\right)}{V\left(E\right)}\right)}^{m-i}\hfill & forV\left(E\right)>0,\hfill \\ 0\hfill & forV\left(E\right)=0.\hfill \end{array}\right.\hfill \end{array}$$

(2)

When $V\left(E\right)>0$, the equality in (2) holds if and only if E is a $K$-ball introduced in [11] up to the difference of a measure zero set.

For the application of the sharp estimate in (2), the dual polynomial log-Minkowski inequality is established in [11]:

$$\sum _{i=0}^m\frac{n^{m-i}m!}{(m-i)!}{\int }_{{\mathbb{S}}^{n-1}}{\left(log\frac{{\rho }_K\left(u\right)}{{\rho }_L\left(u\right)}\right)}^{m-i}d{V}_L\left(u\right)\le \sum _{i=0}^m\frac{m!}{(m-i)!}{\left(log\frac{V\left(K\right)}{V\left(L\right)}\right)}^{m-i}$$

(3)

where m is an odd number, K, L are two star bodies and $d{V}_L(·)$ is the normalized cone-volume measure

$$d{V}_L(·)=\left(\frac{{\rho }_L^n(·)}{nV\left(L\right)}\right)dS(·).$$

(4)

The equality in (3) holds if and only if there exists $s>0$ such that $K=sL$.

Note that (3) generalizes the dual log-Minkowski inequality for a mixed volume of two star bodies (see [12,16]) and produces the polynomial dual for the conjectured log-Minkowski inequality (see [17]).

In this paper, we study the other extreme value of the anisotropic m-log-potential:

Definition 1.

For $m\in \mathbb{N}$, define

$$W_{log,m}(K,E)=\underset{y\in {\mathbb{R}}^n}{inf}{P}_{log,m}(K,E;y).$$

Note that because $log{\parallel x-y\parallel }_K^{-1}$ may be negative, $W_{log,m}(K,E)$ is defined for integer m.

In Section 2, some fundamental properties of $W_{log,m}(K,E)$ are established. Then, in Section 3, we are able to introduce the new annulus body to solve the problem of optimal estimate for $W_{log,m}(K,E)$ in a precise analytic way. For the application, a polynomial dual log-mixed volume difference law is induced from the optimal estimate.

2. Fundamental Properties

First we recall a metric property in [11] for the Minkowski functional of a star body with respect to the origin.

Proposition 1.

Let $B_2^n$ be the unit ball and

$$\left\{\begin{array}{c}{I}_K=sup\{\tilde{r}\ge 0:\tilde{r}{B}_2^n\subseteq K\},\hfill \\ O_K=inf\{\tilde{r}\ge 0:K\subseteq \tilde{r}{B}_2^n\}.\hfill \end{array}\right.$$

(5)

Then

$$O_K^{-1}\left|x\right|\le {\parallel x\parallel }_K\le I_K^{-1}\left|x\right|forallx\in {\mathbb{R}}^n,$$

(6)

and a quasi-triangle inequality holds for ${\parallel ·\parallel }_K$

$${\parallel x+y\parallel }_K\le I_K^{-1}{O}_K\left({\parallel x\parallel }_K+{\parallel y\parallel }_K\right)forallx,y\in {\mathbb{R}}^n.$$

If m is an even number, the supremum of the anisotropic m-log-potential $V_{log,m}(K,E)\equiv +\infty$ (see [11]). For the infimum of the anisotropic m-log-potential $W_{log,m}(K,E)$, it follows

Proposition 2.

$W_{log,m}(K,E)\equiv -\infty$ for m as an odd number.

Proof. 

Note that K is a star body with respect to the origin and E is a bounded measurable set. Then ${sup}_{x\in E}\left|x\right|<+\infty$. For all $C>0$, let $C_1={\mathrm{e}}^{{\left(\frac{C}{V\left(E\right)}\right)}^{\frac{1}{m}}}>1$, $\left|y\right|>max\left\{2O_K{C}_1,2{sup}_{x\in E}\left|x\right|\right\},$ where $O_K$ is defined in (5). Hence, for all $x\in E$,

$${\parallel x-y\parallel }_K\ge {O_K}^{-1}|x-y|\ge {O_K}^{-1}\left(\right|y|-\left|x\right|)>{O_K}^{-1}\frac{\left|y\right|}{2}>C_1>1.$$

Since m is odd, it follows that

$$\begin{array}{cc}\hfill P_{log,m}(K,E;y)& ={\int }_E{\left(log\frac{1}{{\parallel x-y\parallel }_K}\right)}^{m}dx\hfill \\ \hfill & <{\int }_E{\left(log{C}_1^{-1}\right)}^{m}dx\hfill \\ \hfill & ={\int }_E{\left(log{\mathrm{e}}^{{\left(\frac{-C}{V\left(E\right)}\right)}^{\frac{1}{m}}}\right)}^{m}dx\hfill \\ \hfill & =-C,\hfill \end{array}$$

which implies

$$W_{log,m}(K,E)=-\infty \mathrm{via}W=\underset{y\in {\mathbb{R}}^n}{inf}{P}_{log,m}(K,E;y).$$

□

$W_{log,m}(K,E)$ has the following metric properties for the nontrivial case (m is an even number).

Proposition 3.

Let m be an even number.

(i) 

Monotonicity: let $E_1$ and $E_2$ are bounded measurable sets and $E_1\subseteq E_2$. Then $W_{log,m}(K,E_1)\le W_{log,m}(K,E_2)$.

(ii) 

Translation-invariance: for all $z\in {\mathbb{R}}^n$, let $z+E=\{z+y:y\in E\}$. Then $W_{log,m}(K,z+E)=W_{log,m}(K,E)$.

(iii) 

Homogeneity: for all $s>0$, $W_{log,m}(sK,sE)=s^n{W}_{log,m}(K,E)$.

Proof. 

(i) Since $E_1\subseteq E_2$, then for all $y\in {\mathbb{R}}^n$,

$${\int }_{E_1}{\left(log\frac{1}{{\parallel x-y\parallel }_K}\right)}^{m}dx\le {\int }_{E_2}{\left(log\frac{1}{{\parallel x-y\parallel }_K}\right)}^{m}dx.$$

Hence,

$$\begin{array}{cc}\hfill W_{log,m}(K,E_1)& =\underset{y\in {\mathbb{R}}^n}{inf}{\int }_{E_1}{\left(log\frac{1}{{\parallel x-y\parallel }_K}\right)}^{m}dx\hfill \\ \hfill & \le \underset{y\in {\mathbb{R}}^n}{inf}{\int }_{E_2}{\left(log\frac{1}{{\parallel x-y\parallel }_K}\right)}^{m}dx=W_{log,m}(K,E_2).\hfill \end{array}$$

(ii) For all $z\in {\mathbb{R}}^n$, by changing the variables $x=z+x_1$ and $y=z+y_1$, it follows

$$\begin{array}{cc}\hfill W_{log,m}(K,z+E)& =\underset{y\in {\mathbb{R}}^n}{inf}{\int }_{z+E}{\left(log\frac{1}{{\parallel x-y\parallel }_K}\right)}^{m}dx\hfill \\ \hfill & =\underset{y\in {\mathbb{R}}^n}{inf}{\int }_E{\left(log\frac{1}{\parallel x_1{+z-y\parallel }_K}\right)}^{m}d{x}_1\hfill \\ \hfill & =\underset{y_1\in {\mathbb{R}}^n}{inf}{\int }_E{\left(log\frac{1}{\parallel x_1-y_1{\parallel }_K}\right)}^{m}d{x}_1\hfill \\ \hfill & =W_{log,m}(K,E).\hfill \end{array}$$

(iii) For all $\forall s>0$, by changing the variables $x=s\tilde{x}$ and $y=s\tilde{y}$ and the definition of Minkowski functional in (1), it follows that

$$\begin{array}{cc}\hfill W_{log,m}(sK,sE)& =\underset{y\in {\mathbb{R}}^n}{inf}{\int }_{sE}{\left(log\frac{1}{{\parallel x-y\parallel }_{sK}}\right)}^{m}dx\hfill \\ \hfill & =\underset{s\tilde{y}\in {\mathbb{R}}^n}{inf}{\int }_E{\left(log\frac{1}{\parallel s\tilde{x}-s\tilde{y}{\parallel }_{sK}}\right)}^{m}ds\tilde{x}\hfill \\ \hfill & =\underset{\tilde{y}\in {\mathbb{R}}^n}{inf}{\int }_E{\left(log\frac{1}{\parallel \tilde{x}-\tilde{y}{\parallel }_K}\right)}^{m}ds\tilde{x}\hfill \\ \hfill & =s^n{W}_{log,m}(K,E).\hfill \end{array}$$

□

The continuity of the anisotropic m-log-potential $P_{log,m}(K,E;·)$ has already been proven in [11]. From this, it follows that

Lemma 1.

Let m be an even number. The infimum in

$$W_{log,m}(K,E)=\underset{y\in {\mathbb{R}}^n}{inf}{P}_{log,m}(K,E;y)$$

is achieved at some $y\in {\mathbb{R}}^n$.

Proof. 

We first conclude that

$$\underset{\left|y\right|\to +\infty }{lim}{P}_{log,m}(K,E;y)=+\infty .$$

(7)

Actually, note that E is a bounded measurable set, then ${sup}_{x\in E}\left|x\right|<+\infty$. For all $M_1>0$, let

$$\left|y\right|\ge max\left\{2\underset{x\in E}{sup}\left|x\right|,2O_K{\mathrm{e}}^{{\left(\frac{M_1}{V\left(E\right)}\right)}^{\frac{1}{m}}}\right\},$$

where $O_K$ is defined in (5). It follows from m being an even number and (6) that

$$\begin{array}{cc}\hfill P_{log,m}(K,E;y)& ={\int }_E{\left(log\frac{1}{{\parallel x-y\parallel }_K}\right)}^{m}dx\hfill \\ \hfill & ={\int }_E{\left({log\parallel x-y\parallel }_K\right)}^{m}dx\hfill \\ \hfill & \ge {\int }_E{\left(log|O_K{|}^{-1}|x-y|\right)}^{m}dx\hfill \\ \hfill & \ge {\int }_E{\left(log|O_K{|}^{-1}\left(\right|y|-\left|x\right|)\right)}^{m}dx\hfill \\ \hfill & \ge {\int }_E{\left(log\left(2\right|O_K{\left|\right)}^{-1}\left|y\right|\right)}^{m}dx\hfill \\ \hfill & \ge {\int }_E{\left(log{\mathrm{e}}^{{\left(\frac{M_1}{V\left(E\right)}\right)}^{\frac{1}{m}}}\right)}^{m}dx\hfill \\ \hfill & \ge M_1,\hfill \end{array}$$

which implies that (7) holds.

In the following, we will show that $P_{log,m}(K,E;·)\not\equiv +\infty$. As a matter of fact, for $z\in {\mathbb{R}}^n$ and $\left|z\right|\ge {sup}_{x\in E}\left|x\right|$,

$$\begin{array}{cc}\hfill P_{log,m}(K,E;z)& ={\int }_E{\left(log\frac{1}{{\parallel x-z\parallel }_K}\right)}^{m}dx\hfill \\ \hfill & ={\int }_E{\left({log\parallel x-z\parallel }_K\right)}^{m}dx\hfill \\ \hfill & \le {\int }_E{\left(log{I}_K^{-1}|x-z|\right)}^{m}dx\hfill \\ \hfill & \le {\int }_E{\left(log{I}_K^{-1}\left(\right|z|+\left|x\right|)\right)}^{m}dx\hfill \\ \hfill & \le {\int }_E{\left(log2I_K^{-1}\left|z\right|\right)}^{m}dx\hfill \\ \hfill & ={\left(log2I_K^{-1}\left|z\right|\right)}^{m}V\left(E\right)\hfill \\ \hfill & <+\infty ,\hfill \end{array}$$

where $I_K$ is in (5). Let $M_2={\left(log2I_K^{-1}\left|z\right|\right)}^{m}V\left(E\right)$. Because of (7), there exists $D_1\ge 0$ such that for all $y\in \{y\in {\mathbb{R}}^n:|y|>D_1\}$, $P_{log,m}(K,E;y)>M_2$, which implies that

$$z\in D=\{y\in {\mathbb{R}}^n:|y|\le D_1\}.$$

Since $P_{log,m}(K,E;·)$ is continuous and D is compact, it can attain its minimum at a point $y_0$. Then

$$\begin{array}{c}\hfill P_{log,m}(K,E;y_0)=\underset{y\in D}{inf}{P}_{log,m}(K,E;y)\le P_{log,m}(K,E;z)\le M_2\le \underset{y\in D^c}{inf}{P}_{log,m}(K,E;y),\end{array}$$

which implies

$$P_{log,m}(K,E;y_0)=\underset{y\in {\mathbb{R}}^n}{inf}{P}_{log,m}(K,E;y).$$

□

3. Optimal Estimate and Application

Now we are ready to establish the optimal estimate for the infimum of the anisotropic m-log-potential.

Theorem 1.

Let m be an even number. Then

$$\begin{array}{cc}\hfill W_{log,m}(K,E)& \ge \frac{m!V\left(K\right)}{n^m}\sum _{i=0}^m\frac{1}{(m-i)!}{\left[log\left({\left({\left(\frac{V\left(E\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\frac{V\left(E\right)}{2V\left(K\right)}\right)\right]}^{m-i}\hfill \\ \hfill & \times \left[\left({(-1)}^i-1\right){\left({\left(\frac{V\left(E\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\left({(-1)}^i+1\right)\frac{V\left(E\right)}{2V\left(K\right)}\right],\hfill \end{array}$$

(8)

where the equality holds if and only if E is a K-annulus body with outer radius a and inner radius $\frac{1}{a}$ up to a difference of a measure zero set, namely there exists $y\in {\mathbb{R}}^n$ such that

$$V\left(E\cap {\left(R_a^K\left(y\right)\right)}^c\right)=V\left(R_a^K\left(y\right)\cap E^c\right)=0$$

where $a={\left({\left({\left(\frac{V\left(E\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\frac{V\left(E\right)}{2V\left(K\right)}\right)}^{\frac{1}{n}}$.

Proof. 

Let $y\in {\mathbb{R}}^n$ be fixed, and note that $a={\left({\left({\left(\frac{V\left(E\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\frac{V\left(E\right)}{2V\left(K\right)}\right)}^{\frac{1}{n}}>1$ and

$$0<\frac{1}{a}={\left({\left({\left(\frac{V\left(E\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}-\frac{V\left(E\right)}{2V\left(K\right)}\right)}^{\frac{1}{n}}<1,$$

which imply

$$V\left(R_a^K\left(y\right)\right)=\left[a^n-{\left(\frac{1}{a}\right)}^n\right]V\left(K\right)=V\left(E\right).$$

Note that

$$\begin{array}{cc}\hfill V\left(E\cap {\left(R_a^K\left(y\right)\right)}^c\right)& =V\left(E\backslash R_a^K\left(y\right)\right)\hfill \\ \hfill & =V\left(E\right)-V\left(R_a^K\left(y\right)\cap E\right)\hfill \\ \hfill & =V\left(R_a^K\left(y\right)\right)-V\left(R_a^K\left(y\right)\cap E\right)\hfill \\ \hfill & =V\left(R_a^K\left(y\right)\backslash E\right)\hfill \\ \hfill & =V\left(R_a^K\left(y\right)\cap E^c\right),\hfill \end{array}$$

which, together with the following elementary computations

$$\left\{\begin{array}{c}{\parallel x-y\parallel }_K>a(or<\frac{1}{a})\mathrm{and}{(loga)}^m<{\left({log\parallel x-y\parallel }_K\right)}^m\mathrm{for}\mathrm{all}x\in E\cap {\left(R_a^K\left(y\right)\right)}^c,\hfill \\ \frac{1}{a}\le {\parallel x-y\parallel }_K\le a\mathrm{and}0\le {\left({log\parallel x-y\parallel }_K\right)}^m\le {(loga)}^m\mathrm{for}\mathrm{all}x\in R_a^K\left(y\right)\cap E^c,\hfill \end{array}\right.$$

implies

$$\begin{array}{cc}\hfill {\int }_{R_a^K\left(y\right)\cap E^c}{\left({log\parallel x-y\parallel }_K\right)}^{m}dx& \le {\left(loga\right)}^{m}V\left(R_a^K\left(y\right)\cap E^c\right)\hfill \\ & ={\left(loga\right)}^{m}V\left(E\cap {\left(R_a^K\left(y\right)\right)}^c\right)\hfill \\ & \le {\int }_{E\cap {\left(R_a^K\left(y\right)\right)}^c}{\left({log\parallel x-y\parallel }_K\right)}^{m}dx.\hfill \end{array}$$

(9)

Note that m is an even number, then

$$\begin{array}{cc}\hfill & P_{log,m}(K,E;y)\hfill \\ \hfill & ={\int }_E{\left(log\frac{1}{{\parallel x-y\parallel }_K}\right)}^{m}dx\hfill \\ \hfill & ={\int }_E{\left({log\parallel x-y\parallel }_K\right)}^{m}dx\hfill \\ \hfill & ={\int }_{{\left(R_a^K\left(y\right)\right)}^c\cap E}{\left({log\parallel x-y\parallel }_K\right)}^{m}dx+{\int }_{R_a^K\left(y\right)\cap E}{\left({log\parallel x-y\parallel }_K\right)}^{m}dx\hfill \\ \hfill & \ge {\int }_{R_a^K\left(y\right)\cap E^c}{\left({log\parallel x-y\parallel }_K\right)}^{m}dx+{\int }_{R_a^K\left(y\right)\cap E}{\left({log\parallel x-y\parallel }_K\right)}^{m}dx\hfill \\ \hfill & ={\int }_{R_a^K\left(y\right)}{\left({log\parallel x-y\parallel }_K\right)}^{m}dx.\hfill \\ \hfill & =m{\int }_{\left\{x:\frac{1}{a}\le {\parallel x-y\parallel }_K\le a\right\}}{\int }_1^{{\parallel x-y\parallel }_K}{s}^{-1}{(logs)}^{m-1}dsdx\hfill \\ \hfill & =m{\int }_{\left\{x:1\le {\parallel x-y\parallel }_K\le a\right\}}{\int }_1^{{\parallel x-y\parallel }_K}{s}^{-1}{(logs)}^{m-1}dsdx\hfill \\ \hfill & -m{\int }_{\left\{x:\frac{1}{a}\le {\parallel x-y\parallel }_K\le 1\right\}}{\int }_{{\parallel x-y\parallel }_K}^1{s}^{-1}{(logs)}^{m-1}dsdx\hfill \\ \hfill & :=I_1+I_2.\hfill \end{array}$$

(10)

By Fubini’s theorem, it follows

$$\begin{array}{cc}\hfill I_1& =m{\int }_1^a{s}^{-1}{(logs)}^{m-1}{\int }_{\left\{x:s\le {\parallel x-y\parallel }_K\le a\right\}}dxds\hfill \\ \hfill & =m{\int }_1^a{s}^{-1}{(logs)}^{m-1}\left(a^n-s^n\right)V\left(K\right)ds\hfill \\ \hfill & =mV\left(K\right)a^n{\int }_1^a{s}^{-1}{(logs)}^{m-1}ds-mV\left(K\right){\int }_1^a{s}^{n-1}{(logs)}^{m-1}ds,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill I_2& =-m{\int }_{\frac{1}{a}}^1{s}^{-1}{(logs)}^{m-1}{\int }_{\left\{x:\frac{1}{a}\le {\parallel x-y\parallel }_K\le s\right\}}dxds\hfill \\ \hfill & =-m{\int }_{\frac{1}{a}}^1{s}^{-1}{(logs)}^{m-1}\left(s^n-\frac{1}{a^n}\right)V\left(K\right)ds\hfill \\ \hfill & =-mV\left(K\right){\int }_{\frac{1}{a}}^1{s}^{n-1}{(logs)}^{m-1}ds+\frac{mV\left(K\right)}{a^n}{\int }_{\frac{1}{a}}^1{s}^{-1}{(logs)}^{m-1}ds.\hfill \end{array}$$

Then, by integration by parts, it follows

$$\begin{array}{cc}\hfill & I_1+I_2\hfill \\ \hfill & =mV\left(K\right)\left[\frac{1}{a^n}{\int }_{\frac{1}{a}}^1{s}^{-1}{(logs)}^{m-1}ds+a^n{\int }_1^a{s}^{-1}{(logs)}^{m-1}ds\right.\hfill \\ \hfill & \left.-{\int }_{\frac{1}{a}}^a{s}^{n-1}{(logs)}^{m-1}ds\right]\hfill \\ \hfill & =mV\left(K\right)\left[\frac{1}{m{a}^n}{\left.{(logs)}^m\right|}_{\frac{1}{a}}^1+\frac{a^n}{m}{\left.{(logs)}^m\right|}_1^a\right.\hfill \\ \hfill & \left.{\left.-(m-1)!s^n\sum _{i=1}^m\frac{{(-1)}^{i-1}{(logs)}^{m-i}}{n^i(m-i)!}\right|}_{\frac{1}{a}}^a\right]\hfill \\ \hfill & =m!V\left(K\right)\sum _{i=0}^m\frac{1}{n^i(m-i)!}{(loga)}^{m-i}\left[-{\left(\frac{1}{a}\right)}^n-{(-1)}^{i-1}{a}^n\right]\hfill \\ \hfill & =\frac{m!V\left(K\right)}{n^m}\sum _{i=0}^m\frac{1}{(m-i)!}{\left[log\left({\left({\left(\frac{V\left(E\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\frac{V\left(E\right)}{2V\left(K\right)}\right)\right]}^{m-i}\hfill \\ \hfill & \times \left[\left({(-1)}^i-1\right){\left({\left(\frac{V\left(E\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\left({(-1)}^i+1\right)\frac{V\left(E\right)}{2V\left(K\right)}\right].\hfill \end{array}$$

(11)

Hence, by (10) and (11), it follows that

$$\begin{array}{cc}\hfill W_{log,m}(K,E)& =\underset{y\in {\mathbb{R}}^n}{inf}{P}_{log,m}(K,E;y)\hfill \\ \hfill & =\underset{y\in {\mathbb{R}}^n}{inf}{\int }_E{\left(log\frac{1}{{\parallel x-y\parallel }_K}\right)}^{m}dx\hfill \\ \hfill & =\underset{y\in {\mathbb{R}}^n}{inf}{\int }_E{\left({log\parallel x-y\parallel }_K\right)}^{m}dx\hfill \\ \hfill & \ge \underset{y\in {\mathbb{R}}^n}{inf}{\int }_{R_a^K\left(y\right)}{\left({log\parallel x-y\parallel }_K\right)}^{m}dx\hfill \\ \hfill & =\frac{m!V\left(K\right)}{n^m}\sum _{i=0}^m\frac{1}{(m-i)!}{\left[log\left({\left({\left(\frac{V\left(E\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\frac{V\left(E\right)}{2V\left(K\right)}\right)\right]}^{m-i}\hfill \\ \hfill & \times \left[\left({(-1)}^i-1\right){\left({\left(\frac{V\left(E\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\left({(-1)}^i+1\right)\frac{V\left(E\right)}{2V\left(K\right)}\right].\hfill \end{array}$$

To prove the equality in (8), if E is almost a $K$-annulus body up to a difference of a measure zero set, which means there exists $z_1\in {\mathbb{R}}^n$ and a such that

$$V\left(E\cap {\left(R_a^K\left(z_1\right)\right)}^c\right)=V\left(R_a^K\left(z_1\right)\cap E^c\right)=0,$$

which, together with (9), implies

$${\int }_{R_a^K\left(z_1\right)\cap E^c}{\left(log\parallel x-z_1{\parallel }_K\right)}^{m}dx={\int }_{E\cap {\left(R_a^K\left(z_1\right)\right)}^c}{\left(log\parallel x-z_1{\parallel }_K\right)}^{m}dx=0,$$

and hence

$${\int }_E{\left(log\frac{1}{\parallel x-z_1{\parallel }_K}\right)}^{m}dx={\int }_{R_a^K\left(z_1\right)}{\left(log\frac{1}{\parallel x-z_1{\parallel }_K}\right)}^{m}dx,$$

(12)

from (10).

By (10)–(12), it follows

$$\begin{array}{cc}\hfill P_{log,m}(K,E;z_1)& ={\int }_{R_a^K\left(z_1\right)}{\left(log\frac{1}{\parallel x-z_1{\parallel }_K}\right)}^{m}dx\hfill \\ \hfill & =\frac{m!V\left(K\right)}{n^m}\sum _{i=0}^m\frac{1}{(m-i)!}{\left[log\left({\left({\left(\frac{V\left(E\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\frac{V\left(E\right)}{2V\left(K\right)}\right)\right]}^{m-i}\hfill \\ \hfill & \times \left[\left({(-1)}^i-1\right){\left({\left(\frac{V\left(E\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\left({(-1)}^i+1\right)\frac{V\left(E\right)}{2V\left(K\right)}\right],\hfill \end{array}$$

which means the equality in (8) holds.

On the other hand, by Lemma 1, there exists $z_2\in {\mathbb{R}}^n$, $W_{log,m}(K,E)=P_{log,m}(K,E;z_2).$ If E is not a K-annulus body up to a difference of a measure zero set, it follows

$$\begin{array}{c}\hfill V\left(E\cap R_a^K{\left(z_2\right)}^c\right)\ne 0\mathrm{and}V\left(R_a^K\left(z_2\right)\cap E^c\right)\ne 0.\end{array}$$

Then the following strict inequality holds from (9):

$${\int }_{R_a^K\left(z_2\right)\cap E^c}{\left(log\parallel x-z_2{\parallel }_K\right)}^{m}dx<{\int }_{E\cap {\left(R_a^K\left(z_2\right)\right)}^c}{\left(log\parallel x-z_2{\parallel }_K\right)}^{m}dx,$$

which implies the inequality in (10) is also strict, and hence

$$\begin{array}{cc}\hfill & W_{log,m}(K,E)\hfill \\ \hfill & =P_{log,m}(K,E;z_2)\hfill \\ \hfill & ={\int }_E{\left(log\frac{1}{\parallel x-z_2{\parallel }_K}\right)}^{m}dx\hfill \\ \hfill & ={\int }_E{\left(log\parallel x-z_2{\parallel }_K\right)}^{m}dx\hfill \\ \hfill & >{\int }_{R_a^K\left(z_2\right)}{\left(log\parallel x-z_2{\parallel }_K\right)}^{m}dx\hfill \\ \hfill & =\frac{m!V\left(K\right)}{n^m}\sum _{i=0}^m\frac{1}{(m-i)!}{\left[log\left({\left({\left(\frac{V\left(E\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\frac{V\left(E\right)}{2V\left(K\right)}\right)\right]}^{m-i}\hfill \\ \hfill & \times \left[\left({(-1)}^i-1\right){\left({\left(\frac{V\left(E\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\left({(-1)}^i+1\right)\frac{V\left(E\right)}{2V\left(K\right)}\right],\hfill \end{array}$$

which means, if the equality in (8) holds, E must be almost a K-annulus body up to a difference of a measure zero set. □

Remark 1.

We claim that there is no such upper bound for $W_{log,m}(K,E)$ by using $V\left(K\right)$ and $V\left(E\right)$ as in Theorem 1 when m is an even number.

Proof. 

Actually, let $V\left(E\right)$ be fixed. For all $M>0$, let $E=E_1\bigcup E_2$, where $V\left(E_1\right)=V\left(E_2\right)=2^{-1}V\left(E\right)$ and

$$\mathrm{dist}\{E_1,E_2\}=inf\left\{\right|x_1-x_2\left|\right|x_1\in E_1,x_2\in E_2\}>2O_K{\mathrm{e}}^{{\left(\frac{2M}{V\left(E\right)}\right)}^{\frac{1}{m}}}.$$

Then, for all $y\in {\mathbb{R}}^n$, $\mathrm{dist}\{\left\{y\right\},E_1\}>O_K{\mathrm{e}}^{{\left(\frac{2M}{V\left(E\right)}\right)}^{\frac{1}{m}}}$ or $\mathrm{dist}\{\left\{y\right\},E_2\}>O_K{\mathrm{e}}^{{\left(\frac{2M}{V\left(E\right)}\right)}^{\frac{1}{m}}}$. Without loss of generality, suppose $\mathrm{dist}\{\left\{y\right\},E_1\}>O_K{\mathrm{e}}^{{\left(\frac{2M}{V\left(E\right)}\right)}^{\frac{1}{m}}}$, then, by (6), it follows

$$\begin{array}{cc}\hfill P_{log,m}(K,E;y)& ={\int }_E{\left(log\frac{1}{{\parallel x-y\parallel }_K}\right)}^{m}dx,\hfill \\ \hfill & ={\int }_E{(log\parallel x-y\parallel }_K{)}^{m}dx\hfill \\ \hfill & \ge {\int }_E{\left(log{O}_K^{-1}|x-y|\right)}^{m}dx\hfill \\ \hfill & >{\int }_{E_1}{\left(log{O}_K^{-1}|x-y|\right)}^{m}dx\hfill \\ \hfill & >M,\hfill \end{array}$$

which implies

$$W_{log,m}(K,E)=\underset{y\in {\mathbb{R}}^n}{inf}{P}_{log,m}(K,E;y)\ge M.$$

This completes the proof of the remark. □

The infimum of the anisotropic m-log-potential is closely related with the convex geometry analysis. For this, a polynomial dual log-mixed volume difference law can be deduced from the optimal estimate for $W_{log,m}(K,E)$ in Theorem 1.

Theorem 2.

Let m be an even number, $L_1,L_2,K$ be star bodies with respect to the origin, $L_1\subseteq L_2$, and $d{V}_{L_1}\left(u\right)$, $d{V}_{L_2}\left(u\right)$ be the normalized cone-volume measures defined in (4), then

$$\begin{array}{cc}\hfill & V\left(L_2\right){\int }_{{\mathbb{S}}^{n-1}}\sum _{i=0}^m\frac{m!}{n^i(m-i)!}log{\left(\frac{{\rho }_K\left(u\right)}{{\rho }_{L_2}\left(u\right)}\right)}^{m-i}d{V}_{L_2}\left(u\right)\hfill \\ \hfill & -V\left(L_1\right){\int }_{{\mathbb{S}}^{n-1}}\sum _{i=0}^m\frac{m!}{n^i(m-i)!}log{\left(\frac{{\rho }_K\left(u\right)}{{\rho }_{L_1}\left(u\right)}\right)}^{m-i}d{V}_{L_1}\left(u\right)\ge \hfill \\ \hfill & \frac{m!V\left(K\right)}{n^m}\sum _{i=0}^m\frac{1}{(m-i)!}{\left[log\left({\left({\left(\frac{V\left(L_2\right)-V\left(L_1\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\frac{V\left(L_2\right)-V\left(L_1\right)}{2V\left(K\right)}\right)\right]}^{m-i}\hfill \\ \hfill & \times \left[\left({(-1)}^i-1\right){\left({\left(\frac{V\left(L_2\right)-V\left(L_1\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\left({(-1)}^i+1\right)\frac{V\left(L_2\right)-V\left(L_1\right)}{2V\left(K\right)}\right],\hfill \end{array}$$

(13)

where the equality holds if and only if $L_2\backslash L_1$ is a K-annulus body centered at origin with outer radius a and inner radius $\frac{1}{a}(a>0)$ up to a difference of a measure zero set.

Proof. 

Note that ${\rho }_K^{-1}(·)={\parallel ·\parallel }_K$, then, by changing to the polar coordinates and integration by parts, it follows that

$$\begin{array}{cc}\hfill & P_{log,m}(K,L_2\backslash L_1;0)\hfill \\ \hfill & ={\int }_{L_2\backslash L_1}{\left(log\frac{1}{{\parallel x\parallel }_K}\right)}^{m}dx\hfill \\ \hfill & ={\int }_{L_2}{\left(log\frac{1}{{\parallel x\parallel }_K}\right)}^{m}dx-{\int }_{L_1}{\left(log\frac{1}{{\parallel x\parallel }_K}\right)}^{m}dx\hfill \\ \hfill & ={\int }_{L_2}{\left(log{\rho }_K\left(x\right)\right)}^{m}dx-{\int }_{L_1}{\left(log{\rho }_K\left(x\right)\right)}^{m}dx\hfill \\ \hfill & ={\int }_{{\mathbb{S}}^{n-1}}{\int }_0^{{\rho }_{L_2}\left(u\right)}{s}^{n-1}{\left(log{\rho }_K\left(su\right)\right)}^{m}dsdu\hfill \\ \hfill & -{\int }_{{\mathbb{S}}^{n-1}}{\int }_0^{{\rho }_{L_1}\left(u\right)}{s}^{n-1}{\left(log{\rho }_K\left(su\right)\right)}^{m}dsdu\hfill \\ \hfill & =n^{-1}{\int }_{{\mathbb{S}}^{n-1}}{\int }_0^{{\rho }_{L_2}\left(u\right)}{\left(log\left(s^{-1}{\rho }_K\left(u\right)\right)\right)}^{m}d{s}^{n}du\hfill \\ \hfill & -n^{-1}{\int }_{{\mathbb{S}}^{n-1}}{\int }_0^{{\rho }_{L_1}\left(u\right)}{\left(log\left(s^{-1}{\rho }_K\left(u\right)\right)\right)}^{m}d{s}^{n}du\hfill \\ \hfill & =n^{-1}{\int }_{{\mathbb{S}}^{n-1}}{\rho }_{L_2}{\left(u\right)}^n{\left(log\frac{{\rho }_K\left(u\right)}{{\rho }_L\left(u\right)}\right)}^{m}du\hfill \\ \hfill & +n^{-1}m{\int }_{{\mathbb{S}}^{n-1}}{\int }_0^{{\rho }_{L_2}\left(u\right)}{s}^{n-1}{\left(log\left(s^{-1}{\rho }_K\left(u\right)\right)\right)}^{m-1}dsdu\hfill \\ \hfill & -n^{-1}{\int }_{{\mathbb{S}}^{n-1}}{\rho }_{L_1}{\left(u\right)}^n{\left(log\frac{{\rho }_K\left(u\right)}{{\rho }_L\left(u\right)}\right)}^{m}du\hfill \\ \hfill & -n^{-1}m{\int }_{{\mathbb{S}}^{n-1}}{\int }_0^{{\rho }_{L_1}\left(u\right)}{s}^{n-1}{\left(log\left(s^{-1}{\rho }_K\left(u\right)\right)\right)}^{m-1}dsdu\hfill \\ \hfill & ⋮\hfill \\ \hfill & =V\left(L_2\right){\int }_{{\mathbb{S}}^{n-1}}\sum _{i=0}^m\frac{m!}{n^i(m-i)!}log{\left(\frac{{\rho }_K\left(u\right)}{{\rho }_{L_2}\left(u\right)}\right)}^{m-i}d{V}_{L_2}\left(u\right)\hfill \\ \hfill & -V\left(L_1\right){\int }_{{\mathbb{S}}^{n-1}}\sum _{i=0}^m\frac{m!}{n^i(m-i)!}log{\left(\frac{{\rho }_K\left(u\right)}{{\rho }_{L_1}\left(u\right)}\right)}^{m-i}d{V}_{L_1}\left(u\right),\hfill \end{array}$$

(14)

where $d{V}_{L_1}$ and $d{V}_{L_2}$ are defined as in (4).

By Theorem 1, it follows that

$$\begin{array}{cc}\hfill & P_{log,m}(K,L_2\backslash L_1;0)\hfill \\ \hfill & ={\int }_{L_2\backslash L_1}{\left(log\frac{1}{{\parallel x\parallel }_K}\right)}^{m}dx\hfill \\ \hfill & \ge \underset{y\in {\mathbb{R}}^n}{inf}{\int }_{L_2\backslash L_1}{\left(log\frac{1}{{\parallel x-y\parallel }_K}\right)}^{m}dx\hfill \\ \hfill & \ge \frac{m!V\left(K\right)}{n^m}\sum _{i=0}^m\frac{1}{(m-i)!}{\left[log\left({\left({\left(\frac{V(L_2\backslash L_1)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\frac{V(L_2\backslash L_1)}{2V\left(K\right)}\right)\right]}^{m-i}\hfill \\ \hfill & \times \left[\left({(-1)}^i-1\right){\left({\left(\frac{V(L_2\backslash L_1)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\left({(-1)}^i+1\right)\frac{V(L_2\backslash L_1)}{2V\left(K\right)}\right]\hfill \\ \hfill & =\hfill \\ \hfill & \frac{m!V\left(K\right)}{n^m}\sum _{i=0}^m\frac{1}{(m-i)!}{\left[log\left({\left({\left(\frac{V\left(L_2\right)-V\left(L_1\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\frac{V\left(L_2\right)-V\left(L_1\right)}{2V\left(K\right)}\right)\right]}^{m-i}\hfill \\ \hfill & \times \left[\left({(-1)}^i-1\right){\left({\left(\frac{V\left(L_2\right)-V\left(L_1\right)}{2V\left(K\right)}\right)}^2+1\right)}^{\frac{1}{2}}+\left({(-1)}^i+1\right)\frac{V\left(L_2\right)-V\left(L_1\right)}{2V\left(K\right)}\right],\hfill \end{array}$$

which, together with (14), implies (13) holds with the equality holds if and only if $L_2\backslash L_1$ is a K-annulus body centered at origin with outer radius a and inner radius $\frac{1}{a}(a>0)$ up to a difference of a measure zero set. □

4. Conclusions

Theorem 1 and its Remark 1 complete the systematic study of the optimal upper and lower bounds of the extreme value of the anisotropic m-log-potential on a bounded measurable set (for the part of its supremum, we refer to [11]). Note that the anisotropic m-log-potential extends the classical logarithmic potential two-fold in anisotropic and higher order of m ways. By virtue of the wide development of Riesz potential with its better complement logarithmic potential for the end point case in harmonic analysis including function spaces, mathematical physics and partial differential equations (see [1,2,3,4,5,6]), these optimal estimates can be further applied to these related topics.

On the other hand, Brunn–Minkowski inequality and Minkowski inequality including their dual versions and generalizations are main topics in convex geometry analysis (see [12,13,16,17] and their references). The dual log-Minkowski inequality deals with the optimal estimate for mixed volume of two star bodies (see [12,16]), which exists as the dual version for the conjectured log-Minkowski inequality (see [17]). The polynomial dual log-mixed volume difference law in Theorem 2 deduced from the optimal estimate in Theorem 1, deals with the optimal estimate for the difference of mixed volumes of two star bodies, which is totally new and contributes to these theories.