Stability of Volume Inequalities Associated with L_p Zonoids

Abstract

In this paper, we establish stability versions of the volume inequalities associated with $L_p$ zonoids. These results, particularly for the case $1\le p\le 2$, extend the case $p=1$ previously obtained by Brazitikos and Giannopoulos. As applications, we derive several stability inequalities for $L_p$ isotropic convex bodies and for bodies with minimal p-mean width and minimal $L_p$ surface area.

1. Introduction

We shall work in the Euclidean space ${\mathbb{R}}^n$ with the Euclidean norm $\parallel ·\parallel$. Denote the Euclidean unit ball by $B_2^n=\left\{x\in {\mathbb{R}}^n:\parallel x\parallel \le 1\right\}$ and the Euclidean unit sphere by $S^{n-1}=\left\{x\in {\mathbb{R}}^n:\parallel x\parallel =1\right\}.$ A convex body K in ${\mathbb{R}}^n$ is defined as a nonempty compact convex set with interior points, whose support function $h_K:{\mathbb{R}}^n\to \mathbb{R}$ is defined by

$$h_K\left(x\right)=max\{x·y:y\in K\},$$

(1)

where $x·y$ denotes the usual inner product for $x,y\in {\mathbb{R}}^n$. Its volume (Lebesgue measure) is denoted by $\left|K\right|$. If the origin o belongs to the interior of K, the polar body $K^{*}$ of K is given by

$$K^{*}=\left\{x\in {\mathbb{R}}^n:x·y\le 1\mathrm{for}\mathrm{all}y\in K\right\}.$$

In this paper, we establish stability versions of the volume inequalities associated with $L_p$ zonoids, extending the case $p=1$ previously obtained by Brazitikos and Giannopoulos [1]. For $1\le p\le \infty$ and a Borel measure $\nu$ on $S^{n-1}$ (always assumed to be nonnegative and finite, and whose support set is not contained in any great subsphere $S^{n-2}$ in $S^{n-1}$), define $C_p\left(\nu \right)$ as the symmetric convex body in ${\mathbb{R}}^n$ whose support function is given by, for $x\in {\mathbb{R}}^n$,

$$h_{C_p\left(\nu \right)}\left(x\right)=C_{p,\nu }\left(x\right)={\left({\int }_{S^{n-1}}{|x·u|}^{p}d\nu \left(u\right)\right)}^{{\displaystyle \frac{1}{p}}},1\le p<\infty ,$$

(2)

and the case $p=\infty$ is interpreted as the limit $p\to \infty$,

$$h_{C_{\infty }\left(\nu \right)}\left(x\right)=\underset{u\in \mathrm{supp}\nu }{max}|x·u|,p=\infty .$$

The body $C_p\left(\nu \right)$ is usually called the $L_p$ zonoid introduced by Schneider and Weil [2] (see also [3] (p. 606)). In the classical case $p=1$, the zonoid is defined as a convex body that can be approximated (in the Hausdorff metric) by a sequence of vector additions of finitely many line segments (see, e.g., Section 3.5 of the [3]). Write $C_p^{*}\left(\nu \right)$ for the polar body of $C_p\left(\nu \right)$. Let $p^{*}$ be the Hölder conjugate of $p\ge 1$; i.e., ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{p^{*}}}=1$. Denote by $B_p^n$ the unit ball of ${\mathsf{\ell }}_p^n$-space, and by ${\omega }_n={\pi }^{n/2}/\Gamma (1+n/2)$ the volume of the Euclidean unit ball $B_2^n$. For $1\le p<\infty$, let

$$c_{n,p}={\left({\displaystyle \frac{2{\omega }_{n+p-2}}{{\omega }_n{\omega }_{p-1}}}\right)}^{{\displaystyle \frac{n}{p}}},$$

and $c_{n,\infty }={lim}_{p\to \infty }{c}_{n,p}=1$.

It is well known that a Borel measure $\nu$ on $S^{n-1}$ induces a symmetric positive semi-definite $n\times n$ matrix $M_{\nu }$, defined by

$$M_{\nu }={\int }_{S^{n-1}}u\otimes ud\nu \left(u\right),$$

(3)

where $u\otimes u$ represents the orthogonal projection of rank-one onto the space spanned by the unit vector u. When $M_{\nu }$ coincides with the identity matrix $I_n$ on ${\mathbb{R}}^n$, we say that $\nu$ is an isotropic measure on $S^{n-1}$. The measure $\nu$ is a γ-approximation (the real number $\gamma$ always assumes $\gamma >1$) of an isotropic measure when it satisfies

$$I_n⪯M_{\nu }⪯\gamma I_n,$$

(4)

where $A⪯B$ means that for any symmetric $n\times n$ matrices A and B, $B-A$ is positive semi-definite. Inequality (4) is equivalent to

$${\parallel x\parallel }^2\le (M_{\nu }x·x)\le \gamma {\parallel x\parallel }^2$$

(5)

for all $x\in {\mathbb{R}}^n$.

In [1], based on a generalization of Barthe’s continuous version of the Brascamp–Lieb inequalities, Brazitikos and Giannopoulos established the following stability results for $C_1^{*}\left(\nu \right)$:

$$\nu \left(S^{n-1}\right){\left|C_1^{*}\left(\nu \right)\right|}^{{\displaystyle \frac{1}{n}}}\ge {\displaystyle \frac{n{\omega }_n^{\frac{n+1}{n}}}{2{\omega }_{n-1}}}\ge c,$$

where $c>0$ is an absolute constant. Conversely, if $\nu$ is a $\gamma$-approximation of an isotropic measure on $S^{n-1}$ for some $\gamma >1$, then

$$\nu \left(S^{n-1}\right){\left|C_1^{*}\left(\nu \right)\right|}^{{\displaystyle \frac{1}{n}}}\le 2e\gamma .$$

In this paper, we generalize the stability results in the $L_p$ setting for volume inequalities related to $C_p^{*}\left(\nu \right)$, extending them from the previously mentioned case $p=1$ to $1\le p\le 2$ as follows.

Theorem 1.

For $1\le p\le 2$, if ν is a Borel measure on $S^{n-1}$, then

$$\nu \left(S^{n-1}\right){\left|C_p^{*}\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}\ge c_1,$$

and if ν is a γ-approximation of an isotropic measure on $S^{n-1}$, then

$$\nu \left(S^{n-1}\right){\left|C_p^{*}\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}\le 2e{c}_2\gamma ,$$

where $c_1$ and $c_2$ are absolute constants.

In addition, we prove a broader result related to $C_p^{*}\left(\nu \right)$ and $C_p\left(\nu \right)$ for any $1\le p\le \infty$.

Theorem 2.

If ν is a Borel measure on $S^{n-1}$, then for $1\le p\le \infty$,

$$\nu \left(S^{n-1}\right)|C_p^{*}\left(\nu \right){|}^{{\displaystyle \frac{p}{n}}}\ge n{\left({\displaystyle \frac{{\omega }_n}{c_{n,p}}}\right)}^{{\displaystyle \frac{p}{n}}}\mathrm{and}{\displaystyle \frac{n^p}{\nu \left(S^{n-1}\right)}}{\left|C_p\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}\le n^{p-1}{\left({\omega }_n{c}_{n,p}\right)}^{{\displaystyle \frac{p}{n}}}.$$

(6)

Conversely, if ν is a γ-approximation of an isotropic measure on $S^{n-1}$, then

$$\nu \left(S^{n-1}\right){\left|C_p^{*}\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}\le \left\{\begin{array}{cc}{\displaystyle \gamma n|B_p^n{|}^{\frac{p}{n}},}\hfill & 1\le p\le 2,\hfill \\ {\displaystyle {\gamma }^{\frac{p+2}{2}}n{\left|B_p^n\right|}^{\frac{p}{n}},}\hfill & 2<p\le \infty ,\hfill \end{array}\right.$$

(7)

and

$${\displaystyle \frac{n^p}{\nu \left(S^{n-1}\right)}}{\left|C_p\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}\ge {\gamma }^{-{\displaystyle \frac{3p}{2}}}{n}^{p-1}{\left|B_{p^{*}}^n\right|}^{{\displaystyle \frac{p}{n}}}.$$

(8)

When $\nu$ is an isotropic measure on $S^{n-1}$, Theorem 2 was already obtained by Lutwak, Yang, and Zhang [4], and the inequalities in Theorem 2 are all sharp. That is, if p is not an even integer in $[1,\infty )$, then there is equality in either of inequalities (6) if and only if $\nu$ is the normalized Lebesgue measure. For $p\ne 2$, equality holds in (7) or (8) if and only if $\nu$ is a cross measure. Here the cross measure is an even isotropic discrete measure concentrated on $\pm u_1,\dots ,\pm u_n$, where $u_1,\dots ,u_n$ denotes an orthonormal basis of ${\mathbb{R}}^n$.

The Stirling formula yields the following upper and lower bounds depending only on p.

Theorem 3.

Let $1\le p\le \infty$. If ν is a Borel measure on $S^{n-1}$, then

$$\nu \left(S^{n-1}\right)|C_p^{*}\left(\nu \right){|}^{{\displaystyle \frac{p}{n}}}\ge a_p\mathrm{and}{\displaystyle \frac{n^p}{\nu \left(S^{n-1}\right)}}{\left|C_p\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}\le b_p.$$

Conversely, if ν is a γ-approximation of an isotropic measure on $S^{n-1}$, then

$$\nu \left(S^{n-1}\right){\left|C_p^{*}\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}\le \left\{\begin{array}{cc}{\displaystyle \gamma {\alpha }_p,}\hfill & 1\le p\le 2,\hfill \\ {\displaystyle {\gamma }^{\frac{p+2}{2}}{\alpha }_p,}\hfill & 2<p\le \infty ,\hfill \end{array}\right.$$

(9)

and

$${\displaystyle \frac{n^p}{\nu \left(S^{n-1}\right)}}{\left|C_p\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}\ge {\gamma }^{-{\displaystyle \frac{3p}{2}}}{\beta }_p.$$

Here, $a_p,b_p,{\alpha }_p,{\beta }_p$ are constants depending only on p.

In particular, for $1\le p\le 2$, we further get $a_p\ge c_1$ and ${\alpha }_p\le 2e{c}_2$. Hence, Theorem 1 immediately follows.

We observe from (9) that the exponent in the $\gamma$-dependent upper bound differs between the ranges $1\le p\le 2$ and $2<p\le \infty$. This dichotomy is actually a natural geometric phenomenon; for example, the inclusion relations given in (15) and (16) in the $L_p$ setting. A similar phenomenon arises in the context of stability for the minimal $L_p$ surface area. Such stability is known only for $1\le p\le 2$, as shown in [5] (see Theorem 9). This work in [5] builds on the earlier case $p=1$ obtained by Giannopoulos and Papadimitrakis [6] (an alternative proof is given in [1]).

The upper bound of $\nu \left(S^{n-1}\right){\left|C_p^{*}\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}$ for $2<p\le \infty$ in terms of ${\gamma }^{{\displaystyle \frac{p+2}{2}}}$ is demonstrably weaker than the case $1\le p\le 2$. The exponent $(p+2)/2$ arises from two key tools in our proof for $2<p\le \infty$: the continuous Brascamp–Lieb inequality for approximate isotropic measures (23) and the estimate $\nu \left(S^{n-1}\right)\le \gamma n$ from (20). The stability, though weaker, is still explicit and quantifiable in applications where $\gamma$ is close to 1 (i.e., for measures very close to isotropic).

To establish the lower bound for $|C_p\left(\nu \right)|$, we shall use a direct approach due to Lutwak, Yang, and Zhang [4]. This, in turn, uses ideas by Ball [7] and Barthe [8]. More precisely, we apply the Ball–Barthe inequality and mass transportation techniques. The approach we choose is self-contained and elementary. This method has subsequently found extensive applications; see, for example, [9,10,11,12,13,14,15,16,17], among others.

The paper is structured in the following way. Section 2 collects the necessary background materials. Section 3 contains the proofs of the main results, using the continuous Brascamp–Lieb inequality for general Borel measures and for approximate isotropic measures. Moreover, a direct method from Lutwak, Yang, and Zhang [4] will also be used. Section 4 presents some applications, including stability inequalities for $L_p$ isotropic convex bodies and for the bodies with minimal p-mean width and minimal $L_p$ surface area.

2. Background Materials

In this section, we list some basic facts about convex bodies. The books by Schneider [3] and Gardner [18] are good references.

We denote by ${\mathcal{K}}_o^n$ the class of convex bodies in ${\mathbb{R}}^n$ containing the origin in their interiors. For $K\in {\mathcal{K}}_o^n$, its Minkowski functional ${\parallel x\parallel }_K$ is defined by

$${\parallel x\parallel }_K=min\{t>0:x\in tK\},$$

(10)

for $x\in {\mathbb{R}}^n$. Clearly,

$${\parallel ·\parallel }_K=h_{K^{*}}(·).$$

(11)

For each $p\in (0,\infty )$ and $K\in {\mathcal{K}}_o^n$, the volume of K has the integral representation

$$\left|K\right|={\displaystyle \frac{1}{\Gamma (1+\frac{n}{p})}}{\int }_{{\mathbb{R}}^n}{e}^{-{\parallel x\parallel }_K^p}dx,$$

(12)

where $dx$ stands for the Lebesgue measure on ${\mathbb{R}}^n$. The following formula for the volume of K will also be useful.

$$\left|K\right|={\displaystyle \frac{1}{n}}{\int }_{S^{n-1}}{h}_{K^{*}}{\left(u\right)}^{-n}du,$$

(13)

where $du$ is the spherical Lebesgue measure on $S^{n-1}$.

Let ${\left\{e_i\right\}}_{i=1}^n$ be the standard orthonormal basis of ${\mathbb{R}}^n$. The unit ball $B_p^n$ is explicitly given by

$$B_p^n=\{x\in {\mathbb{R}}^n:{\left(\sum _{i=1}^n{|x·e_i|}^p\right)}^{{\displaystyle \frac{1}{p}}}\le 1\},1\le p<\infty ,$$

and

$$B_{\infty }^n=\{x\in {\mathbb{R}}^n:|x · e_i| \le 1,\mathrm{for}\mathrm{all}i=1,\dots ,n\},p=\infty .$$

Applying (12) yields

$$|B_p^n|={\displaystyle \frac{{(2\Gamma (1+\frac{1}{p}))}^n}{\Gamma (1+\frac{n}{p})}}\mathrm{and}\left|B_{\infty }^n\right|=2^n.$$

(14)

A fundamental inclusion relation states that (see, e.g., [4,7])

$$n^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}}{B}_2^n\subseteq B_p^n\subseteq B_2^n,p\in [1,2],$$

(15)

$$B_2^n\subseteq B_p^n\subseteq n^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}}{B}_2^n,p\in (2,\infty ].$$

(16)

For each $x\in {\mathbb{R}}^n$, Equation (3) is equivalent to

$$(M_{\nu }x·x)={\int }_{S^{n-1}}{|x·u|}^{2}d\nu \left(u\right),$$

(17)

and taking the trace yields

$$\mathrm{tr}\left(M_{\nu }\right)=\nu \left(S^{n-1}\right).$$

(18)

Observe that

$$\begin{array}{cc}\hfill {\int }_{S^{n-1}}(M_{\nu }^{-1}u·u)d\nu \left(u\right)& ={\int }_{S^{n-1}}\mathrm{tr}\left(M_{\nu }^{-1}(u\otimes u)\right)d\nu \left(u\right)=\mathrm{tr}\left[M_{\nu }^{-1}\left({\int }_{S^{n-1}}u\otimes ud\nu \left(u\right)\right)\right]\hfill \\ \hfill & =\mathrm{tr}\left(M_{\nu }^{-1}{M}_{\nu }\right)=\mathrm{tr}\left(I_n\right)=n.\hfill \end{array}$$

(19)

If $\nu$ is a $\gamma$-approximation of an isotropic measure on $S^{n-1}$, i.e., ${\gamma }^{-1}\le (M_{\nu }^{-1}u·u)\le 1$ for all $u\in S^{n-1}$, then

$$n={\int }_{S^{n-1}}(M_{\nu }^{-1}u·u)d\nu \left(u\right)\ge {\gamma }^{-1}\nu \left(S^{n-1}\right),$$

and

$$n={\int }_{S^{n-1}}(M_{\nu }^{-1}u·u)d\nu \left(u\right)\le \nu \left(S^{n-1}\right).$$

Therefore

$$n\le \nu \left(S^{n-1}\right)\le \gamma n.$$

(20)

In particular, if $\nu$ is an isotropic measure on $S^{n-1}$, then

$${\parallel x\parallel }^2={\int }_{S^{n-1}}{|x·u|}^{2}d\nu \left(u\right)\mathrm{and}\nu \left(S^{n-1}\right)=n.$$

Based on Barthe’s argument [19], two continuous versions of the Brascamp–Lieb inequalities for general Borel measures and for approximate isotropic measures were established by Brazitikos and Giannopoulos [1].

Lemma 1.

Let ν be a Borel measure on $S^{n-1}$, and let $\left(f_u\right),u\in S^{n-1}$ be a family of functions $f_u:\mathbb{R}\to [0,\infty )$ that satisfy the following two conditions:

(i) There exists a continuous function $F:S^{n-1}\times \mathbb{R}\to (0,\infty )$ and two functions $a,b$ on $S^{n-1}$ with $a<b$ such that for all $(u,t)\in S^{n-1}\times \mathbb{R}$, $f_u\left(t\right)={\mathsf{1}}_{a\left(u\right)\le t\le b\left(u\right)}F(u,t)$.

(ii) There exists a function $U\in L_1\left(\mathbb{R}\right)\cap L_{\infty }\left(\mathbb{R}\right)$ such that $0\le f_u\le U$ for all $u\in S^{n-1}$. Let

$$A_{\nu }=\sqrt{det{M}_{\nu }}exp\left({\int }_{S^{n-1}}log\sqrt{M_{\nu }^{-1}u·u}(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right).$$

(21)

Then

$$\begin{array}{cc}\hfill & A_{\nu }{\int }_{{\mathbb{R}}^n}exp\left\{{\int }_{S^{n-1}}log{f}_u(x·u)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}dx\hfill \\ & \le exp\left\{{\int }_{S^{n-1}}log\left({\int }_{\mathbb{R}}{f}_u\right)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}.\hfill \end{array}$$

(22)

If ν is a γ-approximation of an isotropic measure on $S^{n-1}$, then for a family of functions $f_u:\mathbb{R}\to [0,\infty )$ satisfying conditions (i) and (ii),

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^n}exp\left\{{\int }_{S^{n-1}}log{f}_u(x·u)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}dx\hfill \\ & \le {\gamma }^{{\displaystyle \frac{n}{2}}}exp\left\{{\int }_{S^{n-1}}log\left({\int }_{\mathbb{R}}{f}_u\right)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}.\hfill \end{array}$$

(23)

Brazitikos and Giannopoulos [1] also established the following continuous analogue of the Ball–Barthe inequality for general measures. This extends the case of isotropic measures proved by Lutwak, Yang, and Zhang [4], and the discrete case due to Ball [7] and Barthe [8] (Proposition 9).

Lemma 2.

Let ν be a Borel measure on $S^{n-1}$. For every continuous function $t:\mathrm{supp}\left(\nu \right)\to (0,\infty )$, one has

$$det\left({\int }_{S^{n-1}}t\left(u\right)u\otimes ud\nu \left(u\right)\right)\ge det\left(M_{\nu }\right)exp\left\{{\int }_{S^{n-1}}logt\left(u\right)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}.$$

A simple application of the mean value theorem yields the following standard result in classic analysis.

Lemma 3.

Suppose that $A\subseteq {\mathbb{R}}^n$ is an open convex set and $T:A\to {\mathbb{R}}^n$ is differentiable. If the Jacobian matrix $dT\left(x\right)$ is positive definite for all $x\in A$, then $T:A\to {\mathbb{R}}^n$ is injective.

3. Proofs of the Main Results

It follows from (11) that for $x\in {\mathbb{R}}^n$,

$${\parallel x\parallel }_{C_p^{*}\left(\nu \right)}={\left({\int }_{S^{n-1}}{|x·u|}^{p}d\nu \left(u\right)\right)}^{{\displaystyle \frac{1}{p}}},1\le p<\infty ,$$

(24)

and

$${\parallel x\parallel }_{C_{\infty }^{*}\left(\nu \right)}=\underset{u\in \mathrm{supp}\nu }{max}|x·u|,p=\infty .$$

We now extend the result of Brazitikos and Giannopoulos [1] by proving an upper bound for $\nu \left(S^{n-1}\right){\left|C_p^{*}\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}$ that holds for $1\le p\le 2$.

Theorem 4.

Suppose $1\le p\le 2$. If ν is a γ-approximation of an isotropic measure on $S^{n-1}$, then

$$\nu \left(S^{n-1}\right)|C_p^{*}\left(\nu \right){|}^{{\displaystyle \frac{p}{n}}}\le \gamma n{\left|B_p^n\right|}^{{\displaystyle \frac{p}{n}}}.$$

Proof. 

Since $\nu$ is a $\gamma$-approximation of an isotropic measure on $S^{n-1}$,

$${\gamma }^{-1}\le M_{\nu }^{-1}u·u\le 1$$

(25)

holds for all $u\in S^{n-1}$. By (12), (24), (22) with $f_u\left(t\right)=exp\left\{-{\displaystyle \frac{{\left|t\right|}^p}{M_{\nu }^{-1}u·u}}\right\}$, and (19), we have, for $1\le p\le 2$,

$$\begin{array}{cc}\hfill A_{\nu }\Gamma \left(1+{\displaystyle \frac{n}{p}}\right)\left|C_p^{*}\left(\nu \right)\right|& =A_{\nu }{\int }_{{\mathbb{R}}^n}{e}^{-{\parallel x\parallel }_{C_p^{*}\left(\nu \right)}^p}dx=A_{\nu }{\int }_{{\mathbb{R}}^n}exp\left\{-{\int }_{S^{n-1}}{|x·u|}^{p}d\nu \left(u\right)\right\}dx\hfill \\ \hfill & =A_{\nu }{\int }_{{\mathbb{R}}^n}exp\left\{{\int }_{S^{n-1}}log\left(f_u(x·u)\right)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}dx\hfill \\ \hfill & \le exp\left\{{\int }_{S^{n-1}}log\left({\int }_{\mathbb{R}}{f}_u\right)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}\hfill \\ \hfill & =exp\{{\int }_{S^{n-1}}log(2\Gamma (1+{\displaystyle \frac{1}{p}}){(M_{\nu }^{-1}u·u)}^{{\displaystyle \frac{1}{p}}}\left)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}\hfill \\ \hfill & ={\left(2\Gamma (1+{\displaystyle \frac{1}{p}})\right)}^{n}exp\left\{{\int }_{S^{n-1}}{\displaystyle \frac{1}{p}}log(M_{\nu }^{-1}u·u)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}.\hfill \end{array}$$

This, together with the definition of $A_{\nu }$ (21), gives

$$\begin{array}{cc}\hfill & \sqrt{det{M}_{\nu }}\Gamma \left(1+{\displaystyle \frac{n}{p}}\right)\left|C_p^{*}\left(\nu \right)\right|\hfill \\ & \le {\left(2\Gamma (1+{\displaystyle \frac{1}{p}})\right)}^{n}exp\left\{{\int }_{S^{n-1}}({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2}})log(M_{\nu }^{-1}u·u)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}.\hfill \end{array}$$

Obviously, $det{M}_{\nu }\ge 1$ and for $1\le p\le 2$, $({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2}})log\left(M_{\nu }^{-1}u·u\right)\le 0$ for all $u\in S^{n-1}$. Then

$$\left|C_p^{*}\left(\nu \right)\right|\le \left|B_p^n\right|.$$

(26)

Finally, by the fact that $\nu \left(S^{n-1}\right)\le \gamma n$ (see (20)), we conclude that for $1\le p\le 2$,

$$\nu \left(S^{n-1}\right)|C_p^{*}\left(\nu \right){|}^{{\displaystyle \frac{p}{n}}}\le \gamma n{\left|B_p^n\right|}^{{\displaystyle \frac{p}{n}}},$$

as desired. □

For $2<p\le \infty$, the corresponding upper bound for $\nu \left(S^{n-1}\right){\left|C_p^{*}\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}$ is provided by the following theorem.

Theorem 5.

Suppose $2<p\le \infty$. If ν is a γ-approximation of an isotropic measure on $S^{n-1}$, then

$$\nu \left(S^{n-1}\right)|C_p^{*}\left(\nu \right){|}^{{\displaystyle \frac{p}{n}}}\le {\gamma }^{{\displaystyle \frac{p+2}{2}}}n{\left|B_p^n\right|}^{{\displaystyle \frac{p}{n}}}.$$

Proof. 

By (12), (24), (25), (23) with $f_u\left(t\right)=e^{-{\left|t\right|}^p}$, and (19), we get

$$\begin{array}{cc}\hfill \Gamma (1+{\displaystyle \frac{n}{p}})\left|C_p^{*}\left(\nu \right)\right|& ={\int }_{{\mathbb{R}}^n}{e}^{-{\parallel x\parallel }_{C_p^{*}\left(\nu \right)}^p}dx={\int }_{{\mathbb{R}}^n}exp\left\{-{\int }_{S^{n-1}}{|x·u|}^{p}d\nu \left(u\right)\right\}dx\hfill \\ \hfill & \le {\int }_{{\mathbb{R}}^n}exp\left\{-{\int }_{S^{n-1}}{|x·u|}^p(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}dx\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^n}exp\left\{{\int }_{S^{n-1}}log\left(f_u(x·u)\right)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}dx\hfill \\ \hfill & \le {\gamma }^{{\displaystyle \frac{n}{2}}}exp\left\{{\int }_{S^{n-1}}log\left({\int }_{\mathbb{R}}{f}_u\right)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}\hfill \\ \hfill & ={\gamma }^{{\displaystyle \frac{n}{2}}}exp\left\{{\int }_{S^{n-1}}log\left(2\Gamma (1+{\displaystyle \frac{1}{p}})\right)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}\hfill \\ \hfill & ={\gamma }^{{\displaystyle \frac{n}{2}}}{\left(2\Gamma (1+{\displaystyle \frac{1}{p}})\right)}^n.\hfill \end{array}$$

This gives that

$$\left|C_p^{*}\left(\nu \right)\right|\le {\gamma }^{{\displaystyle \frac{n}{2}}}\left|B_p^n\right|.$$

(27)

Similarly, for $p=\infty$, applying (23) with $f_u\left(t\right)={\mathbf{1}}_{[-1,1]}\left(t\right)$ yields

$$\left|C_{\infty }^{*}\left(\nu \right)\right|\le {\gamma }^{{\displaystyle \frac{n}{2}}}{2}^n.$$

(28)

Therefore, using $\nu \left(S^{n-1}\right)\le \gamma n$, we obtain for $2<p\le \infty$ that

$$\nu \left(S^{n-1}\right)|C_p^{*}\left(\nu \right){|}^{{\displaystyle \frac{p}{n}}}\le {\gamma }^{{\displaystyle \frac{p+2}{2}}}n{\left|B_p^n\right|}^{{\displaystyle \frac{p}{n}}},$$

as desired. □

Next, we use a direct approach due to Lutwak, Yang, and Zhang [4] to establish the lower bound for $|C_p\left(\nu \right)|$. First, we reproduce a lemma, whose proof is similar to that in [4]. For completeness, we give the details.

Lemma 4 ([4]).

Suppose that $1\le p\le \infty$ and ν is a γ-approximation of an isotropic measure on $S^{n-1}$. If $t\in L_{p^{*}}\left(\nu \right)$, then

$$\parallel {\int }_{S^{n-1}}ut\left(u\right)d\nu \left(u\right){\parallel }_{C_p\left(\nu \right)}^{p^{*}}\le \gamma {\int }_{S^{n-1}}{\left|t\left(u\right)\right|}^{p^{*}}(M_{\nu }^{-1}u·u)d\nu \left(u\right).$$

Proof. 

Let $y={\int }_{S^{n-1}}ut\left(u\right)d\nu \left(u\right)$. Recall that ${\gamma }^{-1}\le M_{\nu }^{-1}u·u\le 1$ for all $u\in S^{n-1}$. Define $M_p$ as the closure

$$M_p=\mathrm{cl}\{y\in {\mathbb{R}}^n:{\left({\int }_{S^{n-1}}{\left|t\left(u\right)\right|}^{p^{*}}(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right)}^{{\displaystyle \frac{1}{p^{*}}}}\le 1\},1\le p<\infty ,$$

(29)

and

$$M_{\infty }=\mathrm{cl}\left\{y\in {\mathbb{R}}^n:{\int }_{S^{n-1}}\left|t\left(u\right)\right|(M_{\nu }^{-1}u·u)d\nu \left(u\right)\le 1\right\},p=\infty .$$

(30)

Obviously, $M_p$ is a convex body in ${\mathbb{R}}^n$ for all $1\le p\le \infty$.

For $1<p<\infty$, by (1), (29), the Hölder inequality, and the fact that ${\gamma }^{-1}\le M_{\nu }^{-1}u·u$ for all $u\in S^{n-1}$, we have, for $x\in {\mathbb{R}}^n$,

$$\begin{array}{cc}\hfill h_{M_p}\left(x\right)& =\underset{({\int }_{S^{n-1}}{\left|t\left(u\right)\right|}^{p^{*}}(M_{\nu }^{-1}u·u)d\nu \left(u\right){)}^{1/p^{*}}\le 1}{sup}|x·y|\hfill \\ \hfill & =\underset{({\int }_{S^{n-1}}{\left|t\left(u\right)\right|}^{p^{*}}(M_{\nu }^{-1}u·u)d\nu \left(u\right){)}^{1/p^{*}}\le 1}{sup}\left|{\int }_{S^{n-1}}(x·u)t\left(u\right)d\nu \left(u\right)\right|\hfill \\ \hfill & \le \underset{({\int }_{S^{n-1}}{\left|t\left(u\right)\right|}^{p^{*}}(M_{\nu }^{-1}u·u)d\nu \left(u\right){)}^{1/p^{*}}\le 1}{sup}{\left({\int }_{S^{n-1}}{|x·u|}^{p}d\nu \left(u\right)\right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{S^{n-1}}{\left|t\left(u\right)\right|}^{p^{*}}d\nu \left(u\right)\right)}^{{\displaystyle \frac{1}{p^{*}}}}\hfill \\ \hfill & \le \underset{({\int }_{S^{n-1}}{\left|t\left(u\right)\right|}^{p^{*}}(M_{\nu }^{-1}u·u)d\nu \left(u\right){)}^{1/p^{*}}\le 1}{sup}{\left({\int }_{S^{n-1}}{|x·u|}^{p}d\nu \left(u\right)\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times {\left({\int }_{S^{n-1}}{\left|t\left(u\right)\right|}^{p^{*}}\gamma (M_{\nu }^{-1}u·u)d\nu \left(u\right)\right)}^{{\displaystyle \frac{1}{p^{*}}}}\hfill \\ & ={\gamma }^{{\displaystyle \frac{1}{p^{*}}}}{\left({\int }_{S^{n-1}}{|x·u|}^{p}d\nu \left(u\right)\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & ={\gamma }^{{\displaystyle \frac{1}{p^{*}}}}{h}_{C_p\left(\nu \right)}\left(x\right).\hfill \end{array}$$

The case $p=1$ is similar.

For $p=\infty$, it follows from (30) that for $x\in {\mathbb{R}}^n$,

$$\begin{array}{cc}\hfill h_{M_{\infty }}\left(x\right)& =\underset{{\int }_{S^{n-1}}\left|t\left(u\right)\right|(M_{\nu }^{-1}u·u)d\nu \left(u\right)\le 1}{sup}\left|{\int }_{S^{n-1}}(x·u)t\left(u\right)d\nu \left(u\right)\right|\hfill \\ \hfill & \le \underset{{\int }_{S^{n-1}}\left|t\left(u\right)\right|(M_{\nu }^{-1}u·u)d\nu \left(u\right)\le 1}{sup}{\int }_{S^{n-1}}|x·u|\left|t\left(u\right)\right|d\nu \left(u\right)\hfill \\ \hfill & \le \underset{{\int }_{S^{n-1}}\left|t\left(u\right)\right|(M_{\nu }^{-1}u·u)d\nu \left(u\right)\le 1}{sup}{\int }_{S^{n-1}}|x·u|\left|t\left(u\right)\right|\gamma (M_{\nu }^{-1}u·u)d\nu \left(u\right)\hfill \\ \hfill & \le \underset{u\in \mathrm{supp}\nu }{sup}\gamma |x·u|\hfill \\ \hfill & =\gamma h_{C_{\infty }\left(\nu \right)}\left(x\right).\hfill \end{array}$$

Thus, for $1\le p\le \infty$, $M_p\subseteq {\gamma }^{{\displaystyle \frac{1}{p^{*}}}}{C}_p\left(\nu \right)$, and consequently,

$${\parallel y\parallel }_{C_p\left(\nu \right)}\le {\gamma }^{{\displaystyle \frac{1}{p^{*}}}}{\parallel y\parallel }_{M_p}={\gamma }^{{\displaystyle \frac{1}{p^{*}}}}{\left({\int }_{S^{n-1}}{\left|t\left(u\right)\right|}^{p^{*}}(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right)}^{{\displaystyle \frac{1}{p^{*}}}},$$

as desired. □

Theorem 6.

Suppose $1\le p\le \infty$. If ν is a γ-approximation of an isotropic measure on $S^{n-1}$, then

$${\displaystyle \frac{n^p}{\nu \left(S^{n-1}\right)}}{\left|C_p\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}\ge {\gamma }^{-{\displaystyle \frac{3p}{2}}}{n}^{p-1}{\left|B_{p^{*}}^n\right|}^{{\displaystyle \frac{p}{n}}}.$$

Proof. 

Case $1<p\le \infty$: Let $\varphi :\mathbb{R}\to \mathbb{R}$ be the strictly increasing function satisfying

$${\displaystyle \frac{1}{\Gamma \left(\frac{3}{2}\right)}}{\int }_{-\infty }^t{e}^{-s^2}ds={\displaystyle \frac{1}{\Gamma (1+\frac{1}{p^{*}})}}{\int }_{-\infty }^{\varphi \left(t\right)}{e}^{-{\left|s\right|}^{p^{*}}}ds.$$

Taking derivatives on both sides with respect to t gives

$$-t^2=log\Gamma \left({\displaystyle \frac{3}{2}}\right)-log\Gamma (1+{\displaystyle \frac{1}{p^{*}}})-{\left|\varphi \left(t\right)\right|}^{p^{*}}+log{\varphi }^{\prime }\left(t\right).$$

(31)

For $x\in {\mathbb{R}}^n$, define $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ as

$$T\left(x\right)={\int }_{S^{n-1}}u\varphi (x·u)d\nu \left(u\right).$$

(32)

Its differential is

$$dT\left(x\right)={\int }_{S^{n-1}}u\otimes u{\varphi }^{\prime }(x·u)d\nu \left(u\right).$$

(33)

Thus, for all $v\in S^{n-1}$,

$$v·dT\left(x\right)v={\int }_{S^{n-1}}{|u·v|}^2{\varphi }^{\prime }(x·u)d\nu \left(u\right).$$

Since $\nu$ is a $\gamma$-approximation of an isotropic measure on $S^{n-1}$, it is not concentrated on a great subsphere of $S^{n-1}$. Hence, the condition ${\varphi }^{\prime }>0$ ensures that the matrix $dT\left(x\right)$ is positive definite. By Lemma 3, the transformation T: ${\mathbb{R}}^n\to {\mathbb{R}}^n$ is globally injective. Applying Lemma 4 with $t\left(u\right)=\varphi (x·u)$ gives

$${\parallel T\left(x\right)\parallel }_{C_p\left(\nu \right)}^{p^{*}}\le \gamma {\int }_{S^{n-1}}{\left|\varphi (x·u)\right|}^{p^{*}}(M_{\nu }^{-1}u·u)d\nu \left(u\right).$$

(34)

By (12), (5), (17), (31) with $t=x·u$, (19), the fact of $det\left(M_{\nu }\right)\ge 1$, Lemma 2 with $t\left(u\right)={\varphi }^{\prime }(x·u)$, (33), (34), letting $y=T\left(x\right)$, and again (12), we obtain

$$\begin{array}{cc}\hfill & {\gamma }^{-{\displaystyle \frac{n}{2}}}\Gamma {\left({\displaystyle \frac{1}{2}}\right)}^n={\int }_{{\mathbb{R}}^n}{e}^{-{\gamma \parallel x\parallel }^2}dx\le {\int }_{{\mathbb{R}}^n}{e}^{-|M_{\nu }x·x|}dx={\int }_{{\mathbb{R}}^n}exp\left\{-{\int }_{S^{n-1}}{|x·u|}^{2}d\nu \left(u\right)\right\}dx\hfill \\ \hfill & \le {\int }_{{\mathbb{R}}^n}exp\left\{-{\int }_{S^{n-1}}{|x·u|}^2(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}dx\hfill \\ \hfill & ={\left({\displaystyle \frac{\Gamma \left(\frac{3}{2}\right)}{\Gamma (1+\frac{1}{p^{*}})}}\right)}^n{\int }_{{\mathbb{R}}^n}exp\left\{{\int }_{S^{n-1}}-{\left|\varphi (x·u)\right|}^{p^{*}}(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}\hfill \\ \hfill & \times exp\left\{{\int }_{S^{n-1}}log\left({\varphi }^{\prime }(x·u)\right)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}dx\hfill \\ \hfill & \le {\left({\displaystyle \frac{\Gamma \left(\frac{3}{2}\right)}{\Gamma (1+\frac{1}{p^{*}})}}\right)}^n{\int }_{{\mathbb{R}}^n}{e}^{-{\gamma }^{-1}{\parallel Tx\parallel }_{C_p\left(\nu \right)}^{p^{*}}}det\left(dT\left(x\right)\right)dx\hfill \\ \hfill & \le {\left({\displaystyle \frac{\Gamma \left(\frac{3}{2}\right)}{\Gamma (1+\frac{1}{p^{*}})}}\right)}^n{\int }_{{\mathbb{R}}^n}{e}^{-{\gamma }^{-1}{\parallel y\parallel }_{C_p\left(\nu \right)}^{p^{*}}}dy\hfill \\ \hfill & ={\left({\displaystyle \frac{\Gamma \left(\frac{3}{2}\right)}{\Gamma (1+\frac{1}{p^{*}})}}\right)}^n{\gamma }^{{\displaystyle \frac{n}{p^{*}}}}\left|C_p\left(\nu \right)\right|\Gamma \left(1+{\displaystyle \frac{n}{p^{*}}}\right).\hfill \end{array}$$

Together with (14), this yields

$${\displaystyle \frac{|C_p\left(\nu \right)|}{|B_{p^{*}}^n|}}={\displaystyle \frac{\Gamma (1+\frac{n}{p^{*}})\left|C_p\left(\nu \right)\right|}{{(2\Gamma (1+\frac{1}{p^{*}}))}^n}}\ge {\gamma }^{-{\displaystyle \frac{n}{2}}-{\displaystyle \frac{n}{p^{*}}}}={\gamma }^{{\displaystyle \frac{n}{p}}(1-{\displaystyle \frac{3p}{2}})}.$$

Case $p=1$: Let $\varphi :\mathbb{R}\to (-1,1)$ be the strictly increasing function satisfying

$${\displaystyle \frac{1}{\Gamma \left(\frac{3}{2}\right)}}{\int }_{-\infty }^t{e}^{-s^2}ds={\int }_{-\infty }^{\varphi \left(t\right)}{\mathbf{1}}_{[-1,1]}\left(s\right)ds.$$

Differentiating both sides with respect to t gives

$$-t^2=log\Gamma \left({\displaystyle \frac{3}{2}}\right)+log{\varphi }^{\prime }\left(t\right).$$

(35)

Define $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ as in (32) with the function $\varphi$ chosen above. In fact, $T:{\mathbb{R}}^n\to C_1\left(\nu \right)$; i.e.,

$$T\left({\mathbb{R}}^n\right)\subseteq C_1\left(\nu \right).$$

(36)

To see this, applying Lemma 4 with $t\left(u\right)=\varphi (x·u)$ and the fact that $\left|\varphi \right|<1$ yields

$${\parallel T\left(x\right)\parallel }_{C_1\left(\nu \right)}\le \underset{u\in S^{n-1}}{max}\left|\varphi (x·u)\right|<1,x\in {\mathbb{R}}^n.$$

This, by (10), gives $T\left(x\right)\in C_1\left(\nu \right)$. The differential of T is

$$dT\left(x\right)={\int }_{S^{n-1}}u\otimes u{\varphi }^{\prime }(x·u)d\nu \left(u\right).$$

(37)

For $x\in {\mathbb{R}}^n$, the condition ${\varphi }^{\prime }>0$ ensures that the matrix $dT\left(x\right)$ is positive definite. By Lemma 3, the transformation T: ${\mathbb{R}}^n\to C_1\left(\nu \right)\subset {\mathbb{R}}^n$ is globally injective.

Now, by (12), (5), (17), (35) with $t=x·u$, (18), the fact of $det\left(M_{\nu }\right)\ge 1$, (37), Lemma 2 with $t\left(u\right)={\varphi }^{\prime }(x·u)$, letting $y=T\left(x\right)$, and (36), we arrive at

$$\begin{array}{cc}\hfill & {\gamma }^{-{\displaystyle \frac{n}{2}}}\Gamma {\left({\displaystyle \frac{1}{2}}\right)}^n={\int }_{{\mathbb{R}}^n}{e}^{-{\gamma \parallel x\parallel }^2}dx\le {\int }_{{\mathbb{R}}^n}{e}^{-|M_{\nu }x·x|}dx={\int }_{{\mathbb{R}}^n}exp\left\{-{\int }_{S^{n-1}}{|x·u|}^{2}d\nu \left(u\right)\right\}dx\hfill \\ \hfill & \le {\int }_{{\mathbb{R}}^n}exp\left\{-{\int }_{S^{n-1}}{|x·u|}^2(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}dx\hfill \\ & =\Gamma {\left({\displaystyle \frac{3}{2}}\right)}^n{\int }_{{\mathbb{R}}^n}exp\left\{{\int }_{S^{n-1}}log\left({\varphi }^{\prime }(x·u)\right)(M_{\nu }^{-1}u·u)d\nu \left(u\right)\right\}dx\hfill \\ \hfill & \le \Gamma {\left({\displaystyle \frac{3}{2}}\right)}^n{\int }_{{\mathbb{R}}^n}det\left(dT\left(x\right)\right)dx\le \Gamma {\left({\displaystyle \frac{3}{2}}\right)}^n{\int }_{C_1\left(\nu \right)}dy=\Gamma {\left({\displaystyle \frac{3}{2}}\right)}^n\left|C_1\left(\nu \right)\right|,\hfill \end{array}$$

and hence,

$$\left|C_1\left(\nu \right)\right|\ge {\gamma }^{-{\displaystyle \frac{n}{2}}}{2}^n.$$

Finally, combining this with $\nu \left(S^{n-1}\right)\le \gamma n$ gives, for $1\le p\le \infty$,

$${\displaystyle \frac{n^p}{\nu \left(S^{n-1}\right)}}{\left|C_p\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}\ge {\gamma }^{-{\displaystyle \frac{3p}{2}}}{n}^{p-1}{\left|B_{p^{*}}^n\right|}^{{\displaystyle \frac{p}{n}}},$$

as desired. □

To finish the proof of Theorem 2, we need to show the following results.

Theorem 7.

Suppose $1\le p\le \infty$. If ν is a Borel measure on $S^{n-1}$, then

$$\nu \left(S^{n-1}\right)|C_p^{*}\left(\nu \right){|}^{{\displaystyle \frac{p}{n}}}\ge {\displaystyle \frac{n{\omega }_{p-1}{\omega }_n^{\frac{n+p}{n}}}{2{\omega }_{n+p-2}}}\mathrm{and}{\displaystyle \frac{n^p}{\nu \left(S^{n-1}\right)}}{\left|C_p\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}\le {\displaystyle \frac{2n^{p-1}{\omega }_{n+p-2}{\omega }_n^{\frac{p-n}{n}}}{{\omega }_{p-1}}}.$$

Proof. 

For $1\le p<\infty$, combining (13), the Hölder inequality, (2), Fubini’s theorem, and the fact that ${\int }_{S^{n-1}}{|u·v|}^{p}du={\displaystyle \frac{2{\omega }_{n+p-2}}{{\omega }_{p-1}}}$ for any $v\in S^{n-1}$, we obtain

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{\left|C_p^{*}\left(\nu \right)\right|}{{\omega }_n}}\right)}^{-{\displaystyle \frac{1}{n}}}& ={\left({\displaystyle \frac{1}{n{\omega }_n}}{\int }_{S^{n-1}}{h}_{C_p\left(\nu \right)}{\left(u\right)}^{-n}du\right)}^{-{\displaystyle \frac{1}{n}}}\le {\left({\displaystyle \frac{1}{n{\omega }_n}}{\int }_{S^{n-1}}{h}_{C_p\left(\nu \right)}{\left(u\right)}^{p}du\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & ={\left({\displaystyle \frac{1}{n{\omega }_n}}{\int }_{S^{n-1}}{\int }_{S^{n-1}}{|u·v|}^{p}dud\nu \left(v\right)\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & ={\left({\displaystyle \frac{2\nu \left(S^{n-1}\right){\omega }_{n+p-2}}{n{\omega }_n{\omega }_{p-1}}}\right)}^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

Thus,

$$\left|C_p^{*}\left(\nu \right)\right|\ge {\omega }_n{\left({\displaystyle \frac{2\nu \left(S^{n-1}\right){\omega }_{n+p-2}}{n{\omega }_n{\omega }_{p-1}}}\right)}^{-{\displaystyle \frac{n}{p}}}.$$

(38)

Notice that $n\le \nu \left(S^{n-1}\right)\le \gamma n$ and

$$\underset{p\to +\infty }{lim}{\left({\displaystyle \frac{2\nu \left(S^{n-1}\right){\omega }_{n+p-2}}{n{\omega }_n{\omega }_{p-1}}}\right)}^{{\displaystyle \frac{n}{p}}}=1.$$

This shows that (38) still holds for $p=\infty$.

For the second inequality and $1\le p<\infty$, by the Urysohn inequality (see, e.g., Schneider [3] (p. 382)), the Hölder inequality, (2), and Fubini’s theorem, we further have

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{\left|C_p\left(\nu \right)\right|}{{\omega }_n}}\right)}^{{\displaystyle \frac{1}{n}}}& \le {\displaystyle \frac{1}{n{\omega }_n}}{\int }_{S^{n-1}}{h}_{C_p\left(\nu \right)}\left(u\right)du\le {\left({\displaystyle \frac{1}{n{\omega }_n}}{\int }_{S^{n-1}}{h}_{C_p\left(\nu \right)}{\left(u\right)}^{p}du\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & ={\left({\displaystyle \frac{1}{n{\omega }_n}}{\int }_{S^{n-1}}{\int }_{S^{n-1}}{|u·v|}^{p}dud\nu \left(v\right)\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & ={\left({\displaystyle \frac{2\nu \left(S^{n-1}\right){\omega }_{n+p-2}}{n{\omega }_n{\omega }_{p-1}}}\right)}^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

The above inequality remains valid for $p=\infty$ by a limiting argument. □

4. Applications

Let $d\sigma$ be the rotation invariant measure on $S^{n-1}$ with total mass 1. If $\nu$ is a $\gamma$-approximation of an isotropic measure on $S^{n-1}$, then it follows from (13), (26), (27), and (28) that

$${\int }_{S^{n-1}}{h}_{C_p\left(\nu \right)}{\left(u\right)}^{-n}d\sigma \left(u\right)={\displaystyle \frac{\left|C_p^{*}\left(\nu \right)\right|}{{\omega }_n}}\le \left\{\begin{array}{cc}{\displaystyle \frac{|B_p^n|}{{\omega }_n},}\hfill & 1\le p\le 2,\hfill \\ {\displaystyle \frac{{\gamma }^{\frac{n}{2}}\left|B_p^n\right|}{{\omega }_n},}\hfill & 2<p\le \infty .\hfill \end{array}\right.$$

By Markov’s inequality, we obtain that, for any $0<\delta <1$, a random vector $u\in S^{n-1}$ satisfies

$$h_{C_p\left(\nu \right)}\left(u\right)\ge \left\{\begin{array}{cc}{\displaystyle \delta {\left(\frac{{\omega }_n}{|B_p^n|}\right)}^{\frac{1}{n}}\ge \delta d_p{n}^{\frac{1}{p}-\frac{1}{2}},}\hfill & 1\le p\le 2,\hfill \\ {\displaystyle \delta {\gamma }^{-\frac{1}{2}}{\left(\frac{{\omega }_n}{|B_p^n|}\right)}^{\frac{1}{n}}\ge \delta {\gamma }^{-\frac{1}{2}}{d}_p{n}^{\frac{1}{p}-\frac{1}{2}},}\hfill & 2<p\le \infty ,\hfill \end{array}\right.$$

with probability greater than $1-{\delta }^n$. Here

$$d_p={\displaystyle \frac{\sqrt{2\pi e}}{2\Gamma (1+1/p)e^{1/p}{p}^{1/p}}}$$

is a constant depending only on p, whose range for $1\le p\le \infty$ is $\left[\sqrt{{\displaystyle \frac{\pi }{2e}}},{\displaystyle \frac{\sqrt{2\pi e}}{2}}\right]$.

On the other hand, for $1\le p\le 2$ and $u\in S^{n-1}$, by the Hölder inequality, (5), and (20), we obtain

$$\begin{array}{cc}\hfill h_{C_p\left(\nu \right)}\left(u\right)& ={\left({\int }_{S^{n-1}}{|u·v|}^{p}d\nu \left(v\right)\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le {\left[{\left({\int }_{S^{n-1}}{|u·v|}^{2}d\nu \left(v\right)\right)}^{{\displaystyle \frac{p}{2}}}\nu {\left(S^{n-1}\right)}^{1-{\displaystyle \frac{p}{2}}}\right]}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le {\gamma }^{{\displaystyle \frac{1}{2}}}\nu {\left(S^{n-1}\right)}^{{\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le {\gamma }^{{\displaystyle \frac{1}{p}}}{n}^{{\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2}}},\hfill \end{array}$$

and from (5),

$$\begin{array}{c}\hfill h_{C_p\left(\nu \right)}\left(u\right)\ge {\left({\int }_{S^{n-1}}{|u·v|}^{2}d\nu \left(v\right)\right)}^{{\displaystyle \frac{1}{p}}}\ge 1.\end{array}$$

For $2<p\le \infty$ the above inequalities are reversed.

Hence, we obtain the following two-sides inequalities.

Theorem 8.

Suppose $1\le p\le \infty$ and $0<\delta <1$. If ν is a γ-approximation of an isotropic measure on $S^{n-1}$, then with probability greater than $1-{\delta }^n$ we have, for a random $u\in S^{n-1}$,

$$\delta c{n}^{{\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2}}}\le h_{C_p\left(\nu \right)}\left(u\right)\le {\gamma }^{{\displaystyle \frac{1}{p}}}{n}^{{\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2}}},1\le p\le 2,$$

and

$$\delta c{\gamma }^{-{\displaystyle \frac{1}{2}}}{n}^{{\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2}}}\le h_{C_p\left(\nu \right)}\left(u\right)\le 1,2<p\le \infty ,$$

where c is an absolute constant.

Taking $\delta ={\displaystyle \frac{1}{2}}$ in Theorem 8 yields the following corollary.

Corollary 1.

Suppose $1\le p\le \infty$. If ν is a γ-approximation of an isotropic measure on $S^{n-1}$, then with probability greater than $1-2^{-n}$ we have, for a random $u\in S^{n-1}$,

$$c{n}^{{\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2}}}\le h_{C_p\left(\nu \right)}\left(u\right)\le {\gamma }^{{\displaystyle \frac{1}{p}}}{n}^{{\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2}}},1\le p\le 2,$$

and

$$c{\gamma }^{-{\displaystyle \frac{1}{2}}}{n}^{{\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2}}}\le h_{C_p\left(\nu \right)}\left(u\right)\le 1,2<p\le \infty ,$$

where c is an absolute constant.

For the case $p=1$, Corollary 1 reduces to

$$c\sqrt{n}\le h_{C_1\left(\nu \right)}\left(u\right)\le \gamma \sqrt{n},$$

with probability greater than $1-2^{-n}$. This result was established by Brazitikos and Giannopoulos [1].

For $p\ge 1$, a convex body $K\in {\mathcal{K}}_o^n$ is called $L_p$ isotropic [20] if, for each $u\in S^{n-1}$,

$${\int }_{S^{n-1}}{|u·v|}^{2}d{S}_p(K,v)={\displaystyle \frac{S_p\left(K\right)}{n}},$$

where $S_p(K,·)=h_K{(·)}^{1-p}S(K,·)$ is the $L_p$ surface area measure of K on $S^{n-1}$ introduced by Lutwak [21,22], and $S(K,·)$ is the classical surface area measure of K. Here $S_p\left(K\right)=S_p(K,S^{n-1})$. The $L_p$ surface area measure gave rise to an embryonic $L_p$ Brunn–Minkowski theory, which has expanded rapidly since then. For further details and a comprehensive bibliography on the topic, see [3] (Chapter 9) and the references therein.

One important convex body in the $L_p$ Brunn–Minkowski theory is the $L_p$ projection body ${\Pi }_{p}K$ of $K\in {\mathcal{K}}_o^n$. This body, introduced by Lutwak, Yang, and Zhang [23], is an origin-symmetric convex body in ${\mathbb{R}}^n$ whose support function is given by

$$h_{{\Pi }_{p}K}\left(u\right)={\left({\int }_{S^{n-1}}{|u·v|}^{p}d{S}_p(K,v)\right)}^{{\displaystyle \frac{1}{p}}},u\in S^{n-1}.$$

The following intertwining property of ${\Pi }_p^{*}$ with linear transformations was established in [23]:

$${\Pi }_p^{*}TK=T{\Pi }_p^{*}K,T\in \mathrm{SL}\left(n\right).$$

(39)

We say a convex body $K\in {\mathcal{K}}_o^n$ is γ-approximate $L_p$ isotropic if $n{S}_p(K,·)/S_p\left(K\right)$ is a $\gamma$-approximation of an isotropic measure. Then we immediately obtain the following stability result for $L_p$ isotropic convex bodies. The case $p=1$ is due to Brazitikos and Giannopoulos [1]. Related results were also obtained by Giannopoulos and Papadimitrakis [6] and Yu [5].

Corollary 2.

If $K\in {\mathcal{K}}_o^n$ is a γ-approximate $L_p$ isotropic convex body, then for a random $u\in S^{n-1}$,

$${\displaystyle \frac{c{\left(S_p\left(K\right)\right)}^{\frac{1}{p}}}{\sqrt{n}}}\le h_{{\Pi }_{p}K}\left(u\right)\le {\displaystyle \frac{{\left(\gamma S_p\left(K\right)\right)}^{\frac{1}{p}}}{\sqrt{n}}},1\le p\le 2,$$

and

$${\displaystyle \frac{c{\left(S_p\left(K\right)\right)}^{\frac{1}{p}}}{\sqrt{n\gamma }}}\le h_{{\Pi }_{p}K}\left(u\right)\le {\left({\displaystyle \frac{S_p\left(K\right)}{n}}\right)}^{{\displaystyle \frac{1}{p}}},2<p<\infty ,$$

with probability greater than $1-2^{-n}$ on $S^{n-1}$.

For $p\ge 1$ and a convex body K in ${\mathbb{R}}^n$, its p-mean width is defined by

$${\omega }_p\left(K\right)=2{\int }_{S^{n-1}}{h}_K{\left(v\right)}^{p}d\sigma \left(v\right).$$

For any $A\in \mathrm{SL}\left(n\right)$, a convex body K is said to have minimal p-mean width if ${\omega }_p\left(K\right)\le {\omega }_p\left(AK\right)$. Yuan, Leng, and Cheung [24] showed that a sufficiently smooth convex body K in ${\mathbb{R}}^n$ has minimal p-mean width if and only if

$${\int }_{S^{n-1}}{|u·v|}^2{h}_K{\left(v\right)}^{p}d\sigma \left(v\right)={\displaystyle \frac{{\omega }_p\left(K\right)}{2n}},$$

for each $u\in S^{n-1}$; i.e., the measure $2n{h}_K{(·)}^{p}d\sigma (·)/{\omega }_p\left(K\right)$ is isotropic on $S^{n-1}$.

Accordingly, we say that a sufficiently smooth convex body K in ${\mathbb{R}}^n$ has γ-approximate minimal p-mean width if the measure $2n{h}_K{(·)}^{p}d\sigma (·)/{\omega }_p\left(K\right)$ is a $\gamma$-approximation of an isotropic measure.

Corollary 3.

If a sufficiently smooth convex body K in ${\mathbb{R}}^n$ has γ-approximate minimal p-mean width, then for a random $u\in S^{n-1}$,

$${\displaystyle \frac{c{\left({\omega }_p\left(K\right)\right)}^{\frac{1}{p}}}{2^{\frac{1}{p}}\sqrt{n}}}\le {\left({\int }_{S^{n-1}}{|u·v|}^p{h}_K{\left(v\right)}^{p}d\sigma \left(v\right)\right)}^{{\displaystyle \frac{1}{p}}}\le {\displaystyle \frac{{\left(\gamma {\omega }_p\left(K\right)\right)}^{\frac{1}{p}}}{2^{\frac{1}{p}}\sqrt{n}}},1\le p\le 2,$$

and

$${\displaystyle \frac{c{\left({\omega }_p\left(K\right)\right)}^{\frac{1}{p}}}{2^{\frac{1}{p}}\sqrt{n\gamma }}}\le {\left({\int }_{S^{n-1}}{|u·v|}^p{h}_K{\left(v\right)}^{p}d\sigma \left(v\right)\right)}^{{\displaystyle \frac{1}{p}}}\le {\left({\displaystyle \frac{{\omega }_p\left(K\right)}{2n}}\right)}^{{\displaystyle \frac{1}{p}}},2<p<\infty ,$$

with probability greater than $1-2^{-n}$ on $S^{n-1}$.

A convex body $K\in {\mathcal{K}}_o^n$ has minimal $L_p$ surface area if $S_p\left(K\right)\le S_p\left(TK\right)$ for every $T\in \mathrm{SL}\left(n\right)$. As another application of Theorem 1, we give an alternative proof of the stability of the minimal $L_p$ surface area established in [5]. The case $p=1$ was obtained by Giannopoulos and Papadimitrakis [6] (see also [1] for an alternative proof).

Theorem 9.

Let $K\in {\mathcal{K}}_o^n$ with volume 1 such that

$$I_n⪯{\displaystyle \frac{1}{\lambda }}{\int }_{S^{n-1}}u\otimes ud{S}_p(K,u)⪯\gamma I_n$$

for some $\gamma >1$ and $\lambda >0$. If $1\le p\le 2$, then

$$S_p\left(TK\right)\le S_p\left(K\right)\le c\gamma S_p\left(TK\right),$$

where c is an absolute constant and $TK$ has the minimal $L_p$ surface area for $T\in \mathrm{SL}\left(n\right)$.

Proof. 

It follows from (24) that $C_p^{*}\left(t\nu \right)=t^{-{\displaystyle \frac{1}{p}}}{C}_p^{*}\left(\nu \right)$. Then

$$\left(t\nu \right)\left(S^{n-1}\right)|C_p^{*}\left(t\nu \right){|}^{{\displaystyle \frac{p}{n}}}=\nu \left(S^{n-1}\right){\left|C_p^{*}\left(\nu \right)\right|}^{{\displaystyle \frac{p}{n}}}$$

(40)

holds for every Borel measure $\nu$ on $S^{n-1}$ and every $t>0$.

Let $\nu (·)=S_p(K,·)$ in (24). For $T\in \mathrm{SL}\left(n\right)$, Theorem 1 applied to the measures $S_p(K,·)$ and $S_p(TK,·)$, along with (40) and (39), yields

$$\begin{array}{cc}\hfill S_p\left(K\right){\left|{\Pi }_p^{*}K\right|}^{{\displaystyle \frac{p}{n}}}& \le 2e{c}_2\gamma \le {\displaystyle \frac{2e{c}_2}{c_1}}\gamma S_p\left(TK\right){\left|{\Pi }_p^{*}\left(TK\right)\right|}^{{\displaystyle \frac{p}{n}}}\hfill \\ \hfill & ={\displaystyle \frac{2e{c}_2}{c_1}}\gamma S_p\left(TK\right){\left|{\Pi }_p^{*}K\right|}^{{\displaystyle \frac{p}{n}}}.\hfill \end{array}$$

Consequently,

$$S_p\left(K\right)\le c\gamma S_p\left(TK\right).$$

The inequality $S_p\left(TK\right)\le S_p\left(K\right)$ follows directly from the minimality of the $L_p$ surface area of $TK$. □