Volume inequalities for $L_0$-Minkowski combination of convex bodies

Abstract

Recently Böröczky, Lutwak, Yang and Zhang proved the $L_0$-Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper extends their results to m ($m\ge 2$) origin-symmetric convex bodies in the plane. Moreover, relying on the recent results of Schuster and Weberndorfer, volume inequalities for $L_0$-Minkowski combination of origin-symmetric convex bodies in ${\mathbb{R}}^n$ and its dual form are established in this paper.

MSC:52A20, 52A40.

1 Introduction

The setting for this article is an Euclidean space ${\mathbb{R}}^n$, $n\ge 2$. A convex body is a compact convex subset of ${\mathbb{R}}^n$ with a non-empty interior. For a compact convex set $K\subset {\mathbb{R}}^n$ and $x\in {\mathbb{R}}^n$, the support function $h_K:{\mathbb{R}}^n\to \mathbb{R}$ is defined by $h_K(x)=max\{x\cdot y:y\in K\}$, where $x\cdot y$ denotes the standard inner product of x and y in ${\mathbb{R}}^n$. The polar body of a convex body K is given by $K^{\ast }=\{x\in {\mathbb{R}}^n:x\cdot y\le 1\text{ for all }y\in K\}$. The Minkowski addition of two convex bodies K and L is defined as $K+L=\{x+y:x\in K,y\in L\}$, and the scalar multiplication λK of K, where $\lambda \ge 0$, is defined as $\lambda K=\{\lambda x:x\in K\}$.

In the early 1960s, Firey [1] extended the Minkowski combination of convex bodies to $L_p$-Minkowski combination for each $p\ge 1$. Furthermore, he established the $L_p$-Brunn-Minkowski inequality which states the following: If $K_i$ ($i=1,2,\dots ,m$) are convex bodies in ${\mathbb{R}}^n$ that contain the origin in their interiors, and ${\lambda }_i\in [0,1]$ satisfying ${\sum }_{i=1}^m{\lambda }_i=1$, then the volumes of the bodies $K_i$ and their $L_p$-Minkowski combination ${\lambda }_1\cdot K_1{+}_p\cdots {+}_p{\lambda }_m\cdot K_m$ are related by

$$V({\lambda }_1\cdot K_1{+}_p\cdots {+}_p{\lambda }_m\cdot K_m)\ge V{(K_1)}^{{\lambda }_1}\cdots V{(K_m)}^{{\lambda }_m},$$

(1.1)

with equality if and only if $K_i$ are equal.

Recently, Böröczky et al. [2] defined the $L_0$-Minkowski combination of convex bodies and proved the $L_0$-Brunn-Minkowski inequality, which is stronger than (1.1), for two origin-symmetric convex bodies in the plane.

Theorem 1.1 [2]

If K and L are origin-symmetric convex bodies in the plane, then for all real $\lambda \in [0,1]$,

$$V((1-\lambda )\cdot K{+}_0\lambda \cdot L)\ge V{(K)}^{1-\lambda }V{(L)}^{\lambda }.$$

When $\lambda \in (0,1)$, equality in the inequality holds if and only if K and L are dilates or K and L are parallelograms with parallel sides.

Our first main result of this paper is to extend Theorem 1.1 to m ($m\ge 2$) origin-symmetric convex bodies in the plane.

Definition If $K_i$ ($i=1,2,\dots ,m$) are convex bodies that contain the origin in their interiors, then for real ${\lambda }_i\ge 0$ (not all zero), the $L_0$-Minkowski combination ${\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m$ of $K_i$ is defined by

$${\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m=\bigcap _{u\in S^{n-1}}\{x\in {\mathbb{R}}^n:x\cdot u\le \prod _{i=1}^m{h}_{K_i}{(u)}^{{\lambda }_i}\}.$$

(1.2)

Theorem 1.2 If $K_i$ ($i=1,2,\dots ,m$) are origin-symmetric convex bodies in the plane, then for all real ${\lambda }_i\in [0,1]$ satisfying ${\sum }_{i=1}^m{\lambda }_i=1$, we have

$$V({\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m)\ge V{(K_1)}^{{\lambda }_1}\cdots V{(K_m)}^{{\lambda }_m}.$$

(1.3)

However, the $L_0$-Brunn-Minkowski inequality in ${\mathbb{R}}^n$ is still an open problem, even for origin-symmetric convex bodies.

Now, with our second main result we focus on the volume estimate for $L_0$-Minkowski combination of origin-symmetric convex bodies in ${\mathbb{R}}^n$.

In [3], Schuster and Weberndorfer established two powerful volume inequalities of the Wulff shape $W_{\nu ,f}$ determined by an f-centered isotropic measure ν (see Section 2 for details). Using their results, we establish the following two inequalities.

Theorem 1.3 If $K_i$ ($i=1,2,\dots ,m$) are origin-symmetric convex bodies in ${\mathbb{R}}^n$, then for all real ${\lambda }_i\in [0,1]$ satisfying ${\sum }_{i=1}^m{\lambda }_i=1$, we have

$$V({\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m)\le \frac{n^{n/2}{(n+1)}^{(n+1)/2}}{n!}\prod _{i=1}^m{(\underset{u\in S^{n-1}}{sup}{h}_{K_i}(u))}^{n{\lambda }_i}.$$

Theorem 1.4 If $K_i$ ($i=1,2,\dots ,m$) are origin-symmetric convex bodies in ${\mathbb{R}}^n$, then for all real ${\lambda }_i\in [0,1]$ satisfying ${\sum }_{i=1}^m{\lambda }_i=1$, we have

$$V\left({({\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m)}^{\ast }\right)\ge \frac{{(n+1)}^{(n+1)/2}}{n!n^{n/2}}\prod _{i=1}^m\underset{u\in S^{n-1}}{inf}\left(h_{K_i}{(u)}^{-n{\lambda }_i}\right).$$

Furthermore, inequalities of mixed volume and normalized $L_0$-mixed volume (given in this paper) for $L_0$-Minkowski combination of not necessarily origin-symmetric convex bodies in ${\mathbb{R}}^n$ are established in the following theorems.

Theorem 1.5 If $K_i$ ($i=1,2,\dots ,m$) and $L_j$ ($j=2,\dots ,n$) are convex bodies in ${\mathbb{R}}^n$ that contain the origin in their interiors, then for all real ${\lambda }_i\in [0,1]$ satisfying ${\sum }_{i=1}^m{\lambda }_i=1$, we have

$$V({\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m,L_2,\dots ,L_n)\le \prod _{i=1}^{m}V{(K_i,L_2,\dots ,L_n)}^{{\lambda }_i}.$$

If K and L are convex bodies in ${\mathbb{R}}^n$ that contain the origin in their interiors, then for $p\ne 0$, the $L_p$-mixed volume $V_p(K,L)$ can be defined as

$$V_p(K,L)=\frac{1}{n}{\int }_{S^{n-1}}{h}_L^p{h}_K^{1-p}d{S}_K={\int }_{S^{n-1}}{\left(\frac{h_L}{h_K}\right)}^{p}d{V}_K,$$

where $S_K$ and $V_K$ are the surface area measure and the cone-volume measure of K, respectively (see Section 2 for the definitions).

The normalized $L_p$-mixed volume ${\overline{V}}_p(K,L)$ is defined by

$${\overline{V}}_p(K,L)={\left(\frac{V_p(K,L)}{V(K)}\right)}^{1/p}={\left({\int }_{S^{n-1}}{\left(\frac{h_L}{h_K}\right)}^{p}d{\overline{V}}_K\right)}^{1/p},$$

where ${\overline{V}}_K$ is the cone-volume probability measure of K (also see Section 2 for the definition).

Note that when p converges to zero, the normalized $L_0$-mixed volume ${\overline{V}}_0(K,L)$ can naturally be given as

$${\overline{V}}_0(K,L)=exp({\int }_{S^{n-1}}log\frac{h_L}{h_K}d{\overline{V}}_K).$$

(1.4)

Theorem 1.6 Suppose that K, L and Q are convex bodies in ${\mathbb{R}}^n$ that contain the origin in their interiors, then for real $\lambda \in [0,1]$, we have

$${\overline{V}}_0(Q,(1-\lambda )\cdot K{+}_0\lambda \cdot L)\le {\overline{V}}_0{(Q,K)}^{1-\lambda }{\overline{V}}_0{(Q,L)}^{\lambda }.$$

Combining the famous variant (proved in [4]) of Aleksandrov’s lemma and the representation of (1.4), we obtain a limit form of ${\overline{V}}_0(K,L)$ in the following theorem.

Theorem 1.7 Suppose that K and L are convex bodies in ${\mathbb{R}}^n$ that contain the origin in their interiors, then we have

$$nV(K)log{\overline{V}}_0(K,L)=\underset{\lambda \to 0}{lim}\frac{V((1-\lambda )\cdot K{+}_0\lambda \cdot L)-V(K)}{\lambda }.$$

The paper is organized as follows. In Section 2 some of the basic notations and preliminaries are provided. Section 3 contains the proofs of the main theorems. Some properties of normalized $L_0$-mixed volume and Wulff shape are discussed in Section 4.

2 Notations and preliminaries

Good general references for the theory of convex bodies are provided by the books [5–9] and the articles [2, 4, 7, 10–25].

The group of nonsingular linear transformations is denoted by $\mathit{GL}(n)$; its members are, in particular, bijections of ${\mathbb{R}}^n$ onto itself. The group of special linear transformations of ${\mathbb{R}}^n$ is denoted by $\mathit{SL}(n)$. These are the members of $\mathit{GL}(n)$ whose determinant is one.

For $\varphi \in \mathit{GL}(n)$, let ${\varphi }^t$, ${\varphi }^{-1}$ and ${\varphi }^{-t}$ denote the transpose, inverse and inverse of the transpose of ϕ, respectively.

For $x\in {\mathbb{R}}^n$, then

$$h_K(\lambda x)=\lambda h_K(x),\text{for }\lambda \ge 0,$$

and

$$h_{\varphi K}(x)=h_K\left({\varphi }^{t}x\right),\text{for }\varphi \in \mathit{GL}(n).$$

Recall that for a Borel set $\omega \subseteq S^{n-1}$, the surface area measure of a convex body K in ${\mathbb{R}}^n$ $S_K(\omega )$ is the $(n-1)$-dimensional Hausdorff measure of the set of all boundary points of K at which there exists a normal vector of K belonging to ω.

Let K be a convex body in ${\mathbb{R}}^n$ that contains the origin in its interior. The cone-volume measure $V_K$ of K is a Borel measure on the unit sphere $S^{n-1}$ defined by

$$d{V}_K=\frac{1}{n}{h}_{K}d{S}_K.$$

(2.1)

Obviously,

$$V(K)=\frac{1}{n}{\int }_{S^{n-1}}{h}_K(u)d{S}_K(u).$$

The cone-volume probability measure ${\overline{V}}_K$ of K is defined by

$${\overline{V}}_K=\frac{1}{V(K)}{V}_K.$$

(2.2)

Let $S^{n-1}$ and B denote the unit sphere centered at the origin and the unit ball in ${\mathbb{R}}^n$, respectively. The n-dimensional volume ${\kappa }_n$ of B and the $(n-1)$-dimensional volume ${\omega }_n$ of $S^{n-1}$ are

$${\kappa }_n=\frac{{\pi }^{\frac{n}{2}}}{\mathrm{\Gamma }(1+\frac{n}{2})},$$

and

$${\omega }_n=n{\kappa }_n.$$

If $K_i$ ($i=1,2,\dots ,n$) are convex bodies in ${\mathbb{R}}^n$, the mixed volume $V(K_1,\dots ,K_n)$ is given by (see [5], [[8], Theorem 5.1.6] or [[26], Section 29])

$$V(K_1,\dots ,K_n)=\frac{1}{n}{\int }_{S^{n-1}}{h}_{K_1}(u)dS(K_2,\dots ,K_n,u),$$

(2.3)

where $S(K_2,\dots ,K_n,\cdot )$ is the mixed area measure of $K_i$ ($i=2,\dots ,n$).

If K is a convex body in ${\mathbb{R}}^n$, the quermassintegrals $W_i(K)$ of K are defined for $0\le i\le n$ by

$$W_i(K)=V(K,n-i;B,i),$$

where the notation $V(K,n-i;B,i)$ signifies that K appears $(n-i)$ times and B appears i times.

The mean width $W(K)$ of a convex body K in ${\mathbb{R}}^n$ is defined by

$$W(K)=\frac{2}{{\omega }_n}{\int }_{S^{n-1}}{h}_K(u)d{\mathcal{H}}^{n-1}(u),$$

where ${\mathcal{H}}^{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure.

It can be shown that

$$\frac{2}{{\kappa }_n}{W}_{n-1}(K)=W(K).$$

(2.4)

Throughout, all Borel measures are understood to be non-negative and finite. We write suppν for the support of a measure ν.

Suppose that ν is a Borel measure on $S^{n-1}$ and f is a positive continuous function on $S^{n-1}$. The Wulff shape $W_{\nu ,f}$ determined by ν and f is defined by

$$W_{\nu ,f}=\{x\in {\mathbb{R}}^n:x\cdot u\le f(u)\text{ for all }u\in supp\nu \}.$$

(2.5)

Obviously,

$$h_{W_{\nu ,f}}(u)\le f(u)\text{for all }u\in supp\nu .$$

(2.6)

Let f be a positive continuous function on $S^{n-1}$. A Borel measure ν on $S^{n-1}$ is called f-centered if

$${\int }_{S^{n-1}}f(u)ud\nu (u)=o.$$

The measure ν is called isotropic if

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

where $u\otimes u$ is the orthogonal projection onto the line spanned by u and $I_n$ denotes the identity map on ${\mathbb{R}}^n$. Thus, ν is isotropic if

$${\int }_{S^{n-1}}{|v\cdot u|}^{2}d\nu (u)=1\text{for all }v\in S^{n-1}.$$

The displacement of $W_{\nu ,f}$ is defined by

$$disp{W}_{\nu ,f}=cd{W}_{\nu ,f}\cdot {\int }_{S^{n-1}}\frac{u}{f(u)}d\nu (u),$$

where $cd{W}_{\nu ,f}$ denotes the centroid of $W_{\nu ,f}$.

3 Proof of main results

The following lemma will be used in the proof of Theorem 1.2.

Lemma 3.1 If $K_i$ ($i=1,2,\dots ,m$) are convex bodies that contain the origin in their interiors, then for real ${\lambda }_i\in (0,1)$ and ${\sum }_{i=1}^m{\lambda }_i=1$, we have

Proof Since $K_i$ ($i=1,2,\dots ,m$) contain the origin in their interiors, thus it is easy to see that the $L_0$-Minkowski combination

$$\frac{{\lambda }_1}{1-{\lambda }_m}\cdot K_1{+}_0\cdots {+}_0\frac{{\lambda }_{m-1}}{1-{\lambda }_m}\cdot K_{m-1}=\bigcap _{u\in S^{n-1}}\{x\in {\mathbb{R}}^n:x\cdot u\le \prod _{i=1}^{m-1}{h}_{K_i}{(u)}^{\frac{{\lambda }_i}{1-{\lambda }_m}}\},$$

(3.1)

also contains the origin in its interior.

Let $x\in (1-{\lambda }_m)\cdot (\frac{{\lambda }_1}{1-{\lambda }_m}\cdot K_1{+}_0\cdots {+}_0\frac{{\lambda }_{m-1}}{1-{\lambda }_m}\cdot K_{m-1}){+}_0{\lambda }_m\cdot K_m$, then combining (2.6) and (3.1) we have

$$\begin{array}{rcl}x\cdot u& \le & h_{\frac{{\lambda }_1}{1-{\lambda }_m}\cdot K_1{+}_0\cdots {+}_0\frac{{\lambda }_{m-1}}{1-{\lambda }_m}\cdot K_{m-1}}{(u)}^{1-{\lambda }_m}{h}_{K_m}{(u)}^{{\lambda }_m}\\ \le & {\left(\prod _{i=1}^{m-1}{h}_{K_i}{(u)}^{\frac{{\lambda }_i}{1-{\lambda }_m}}\right)}^{1-{\lambda }_m}{h}_{K_m}{(u)}^{{\lambda }_m}\\ =& \prod _{i=1}^m{h}_{K_i}{(u)}^{{\lambda }_i}\end{array}$$

for all $u\in S^{n-1}$.

Hence, by (1.2), we have $x\in {\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m$, which yields the lemma directly. □

Proof of Theorem 1.2 We will prove Theorem 1.2 by induction on m.

Obviously, it is true by Theorem 1.1 when $m=2$.

Suppose that the result holds on $(m-1)$. Thus, for real ${\lambda }_i\in (0,1)$ satisfying ${\sum }_{i=1}^m{\lambda }_i=1$, we have

$$V(\frac{{\lambda }_1}{1-{\lambda }_m}\cdot K_1{+}_0\cdots {+}_0\frac{{\lambda }_{m-1}}{1-{\lambda }_m}\cdot K_{m-1})\ge V{(K_1)}^{\frac{{\lambda }_1}{1-{\lambda }_m}}\cdots V{(K_{m-1})}^{\frac{{\lambda }_{m-1}}{1-{\lambda }_m}}.$$

We now consider the situation on m. In fact, since $K_i$ ($i=1,2,\dots ,m$) are origin-symmetric convex bodies in the plane, then by (3.1) we have $\frac{{\lambda }_1}{1-{\lambda }_m}\cdot K_1{+}_0\cdots {+}_0\frac{{\lambda }_{m-1}}{1-{\lambda }_m}\cdot K_{m-1}$ is also an origin-symmetric convex body in the plane.

Then by Lemma 3.1, Theorem 1.1 and the induction hypothesis, we have

Note that Theorem 1.2 also holds when ${\lambda }_m=0$ and ${\lambda }_m=1$, respectively.

Thus Theorem 1.2 holds for all ${\lambda }_i\in [0,1]$ satisfying ${\sum }_{i=1}^m{\lambda }_i=1$. □

In [3], Schuster and Weberndorfer established a sharp bound for the volume of the Wulff shape $W_{\nu ,f}$ determined by an f-centered isotropic measure ν as follows.

Lemma 3.2 [3]

Suppose that f is a positive continuous function on $S^{n-1}$ and that ν is an isotropic f-centered measure. If $disp{W}_{\nu ,f}=0$, then

$$V(W_{\nu ,f})\le \frac{{(n+1)}^{(n+1)/2}}{n!}{\parallel f\parallel }_{L^2(\nu )}^n,$$

with equality if and only if $convsupp\nu$ is a regular simplex inscribed in $S^{n-1}$ and f is constant on suppν.

Proof of Theorem 1.3 Let $d\nu =\frac{1}{{\kappa }_n}du$ and $f(u)={\prod }_{i=1}^m{h}_{K_i}{(u)}^{{\lambda }_i}$ for all $u\in S^{n-1}$ in Lemma 3.2. Obviously, f is a positive continuous function on $S^{n-1}$ and ν is isotropic. By (2.5) we have

$$W_{\nu ,f}=\{x\in {\mathbb{R}}^n:x\cdot u\le \prod _{i=1}^m{h}_{K_i}{(u)}^{{\lambda }_i}\text{ for all }u\in supp\nu \}.$$

Combining it with (1.2), we get

$$W_{\nu ,f}={\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m.$$

(3.2)

Furthermore, since $K_i$ ($i=1,2,\dots ,m$) are origin-symmetric convex bodies, then we have

$$\frac{1}{{\kappa }_n}{\int }_{S^{n-1}}{h}_{K_1}{(u)}^{{\lambda }_1}\cdots h_{K_m}{(u)}^{{\lambda }_m}udu=o,$$

and

$$\frac{1}{{\kappa }_n}{\int }_{S^{n-1}}\frac{u}{h_{K_1}{(u)}^{{\lambda }_1}\cdots h_{K_m}{(u)}^{{\lambda }_m}}du=o.$$

Thus ν is f-centered, and

$$disp{W}_{\nu ,f}=cd{W}_{\nu ,f}\cdot \frac{1}{{\kappa }_n}{\int }_{S^{n-1}}\frac{u}{h_{K_1}{(u)}^{{\lambda }_1}\cdots h_{K_m}{(u)}^{{\lambda }_m}}du=0.$$

Combining (3.2) and Lemma 3.2, we have

$$V({\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m)\le \frac{{(n+1)}^{(n+1)/2}}{n!}{\parallel f\parallel }_{L^2(\nu )}^n.$$

(3.3)

Since ${\lambda }_i\in [0,1]$ satisfying ${\sum }_{i=1}^m{\lambda }_i=1$, then from Hölder’s inequality (see [27]) we have

$$\begin{array}{rcl}{\parallel f\parallel }_{L^2(\nu )}^2& =& \frac{1}{{\kappa }_n}{\int }_{S^{n-1}}f{(u)}^{2}du\\ =& \frac{1}{{\kappa }_n}{\int }_{S^{n-1}}\prod _{i=1}^m{h}_{K_i}{(u)}^{2{\lambda }_i}du\\ \le & n\prod _{i=1}^m{(\underset{u\in S^{n-1}}{sup}{h}_{K_i}(u))}^{2{\lambda }_i}.\end{array}$$

(3.4)

Combining (3.3) and (3.4), we obtain

$$V({\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m)\le \frac{n^{n/2}{(n+1)}^{(n+1)/2}}{n!}\prod _{i=1}^m{(\underset{u\in S^{n-1}}{sup}{h}_{K_i}(u))}^{n{\lambda }_i}.$$

 □

A natural dual to Lemma 3.2 is also given in [3], which provided a sharp lower bound for the volume of the polar of the Wulff shape $W_{\nu ,f}$.

Lemma 3.3 [3]

Suppose that f is a positive continuous function on $S^{n-1}$ and that ν is an isotropic f-centered measure. Then

$$V\left(W_{\nu ,f}^{\ast }\right)\ge \frac{{(n+1)}^{(n+1)/2}}{n!}{\parallel f\parallel }_{L^2(\nu )}^{-n},$$

with equality if and only if $convsupp\nu$ is a regular simplex inscribed in $S^{n-1}$ and f is constant on suppν.

Proof of Theorem 1.4 Let $d\nu =\frac{1}{{\kappa }_n}du$ and $f(u)={\prod }_{i=1}^m{h}_{K_i}{(u)}^{{\lambda }_i}$ for all $u\in S^{n-1}$ in Lemma 3.3. Similarly, from the proof of Theorem 1.3, we know that f is a positive continuous function on $S^{n-1}$ and ν is an isotropic f-centered measure.

Thus, combining (3.2) and Lemma 3.3, we have

$$V\left({({\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m)}^{\ast }\right)\ge \frac{{(n+1)}^{(n+1)/2}}{n!}{\parallel f\parallel }_{L^2(\nu )}^{-n}.$$

(3.5)

Now, combining (3.4) and (3.5), we obtain

$$\begin{array}{rcl}V\left({({\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m)}^{\ast }\right)& \ge & \frac{{(n+1)}^{(n+1)/2}}{n!n^{n/2}}\prod _{i=1}^m{(\underset{u\in S^{n-1}}{sup}{h}_{K_i}(u))}^{-n{\lambda }_i}\\ =& \frac{{(n+1)}^{(n+1)/2}}{n!n^{n/2}}\prod _{i=1}^m\underset{u\in S^{n-1}}{inf}\left(h_{K_i}{(u)}^{-n{\lambda }_i}\right).\end{array}$$

 □

Proof of Theorem 1.5 Since ${\lambda }_i\in [0,1]$ satisfying ${\sum }_{i=1}^m{\lambda }_i=1$, then by (2.3), (2.6), Hölder’s inequality and again (2.3), we have

 □

Letting $L_2=\cdots =L_n={\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m$ in Theorem 1.5 gives the following inequality of mixed volumes.

Corollary 3.4 If $K_i$ ($i=1,2,\dots ,m$) are convex bodies in ${\mathbb{R}}^n$ that contain the origin in their interiors, then for all real ${\lambda }_i\in [0,1]$ satisfying ${\sum }_{i=1}^m{\lambda }_i=1$, we have

$$V({\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m)\le \prod _{i_1=1}^m\cdots \prod _{i_n=1}^{m}V{(K_{i_1},\dots ,K_{i_n})}^{{\lambda }_{i_1}\cdots {\lambda }_{i_n}}.$$

Letting $L_2=\cdots =L_n=B$ in Theorem 1.5 gives the following inequality of quermassintegrals.

Corollary 3.5 If $K_i$ ($i=1,2,\dots ,m$) are convex bodies in ${\mathbb{R}}^n$ that contain the origin in their interiors, then for all real ${\lambda }_i\in [0,1]$ satisfying ${\sum }_{i=1}^m{\lambda }_i=1$, we have

$$W_{n-1}({\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m)\le \prod _{i=1}^m{W}_{n-1}{(K_i)}^{{\lambda }_i}.$$

In view of (2.4), we also obtain the following inequality of mean widths.

Corollary 3.6 If $K_i$ ($i=1,2,\dots ,m$) are convex bodies in ${\mathbb{R}}^n$ that contain the origin in their interiors, then for all real ${\lambda }_i\in [0,1]$ satisfying ${\sum }_{i=1}^m{\lambda }_i=1$, we have

$$W({\lambda }_1\cdot K_1{+}_0\cdots {+}_0{\lambda }_m\cdot K_m)\le \prod _{i=1}^{m}W{(K_i)}^{{\lambda }_i}.$$

Proof of Theorem 1.6 Since K, L and Q are convex bodies in ${\mathbb{R}}^n$ that contain the origin in their interiors, then from (1.4), (1.2), (2.6) and again (1.4), we have

 □

The following variant of Aleksandrov’s lemma (see [[28], p.103]) will be needed in proving Theorem 1.7.

Lemma 3.7 [4]

Suppose that $q_{\lambda }(u)=q(\lambda ,u):I\times S^{n-1}\to (0,\mathrm{\infty })$ is a continuous function, where $I\subset \mathbb{R}$ is an open interval. Suppose also that the convergence in

$$\frac{\partial q(\lambda ,u)}{\partial \lambda }=\underset{\gamma \to 0}{lim}\frac{q(\lambda +\gamma ,u)-q(\lambda ,u)}{\gamma }$$

is uniform on $S^{n-1}$. If ${\{Q_{\lambda }\}}_{\lambda \in I}$ is the family of Wulff shapes associated with $q_{\lambda }$, i.e., for fixed $\lambda \in I$,

$$Q_{\lambda }=\bigcap _{u\in S^{n-1}}\{x\in {\mathbb{R}}^n:x\cdot u\le q(\lambda ,u)\},$$

then

$$\frac{dV(Q_{\lambda })}{d\lambda }={\int }_{S^{n-1}}\frac{\partial q(\lambda ,u)}{\partial \lambda }d{S}_{Q_{\lambda }}(u).$$

Proof of Theorem 1.7 Since K and L are convex bodies in ${\mathbb{R}}^n$ that contain the origin in their interiors, let $q(\lambda ,u)=h_K{(u)}^{1-\lambda }{h}_L{(u)}^{\lambda }$ in Lemma 3.7, then the convergence in

$$\begin{array}{rcl}\frac{\partial (h_K{(u)}^{1-\lambda }{h}_L{(u)}^{\lambda })}{\partial \lambda }& =& \underset{\gamma \to 0}{lim}\frac{h_K{(u)}^{1-(\lambda +\gamma )}{h}_L{(u)}^{(\lambda +\gamma )}-h_K{(u)}^{1-\lambda }{h}_L{(u)}^{\lambda }}{\gamma }\\ =& h_K(u){\left(\frac{h_L(u)}{h_K(u)}\right)}^{\lambda }log\frac{h_L(u)}{h_K(u)}\end{array}$$

is uniform on $S^{n-1}$, and

$$Q_{\lambda }=\bigcap _{u\in S^{n-1}}\{x\in {\mathbb{R}}^n:x\cdot u\le h_K{(u)}^{1-\lambda }{h}_L{(u)}^{\lambda }\}.$$

(3.6)

Observe that $q(0,u)$ is the support function of K, hence $Q_0=K$.

On the one hand, from Lemma 3.7, (2.1), (2.2) and (1.4) we have

$$\begin{array}{rcl}{\frac{dV(Q_{\lambda })}{d\lambda }|}_{\lambda =0}& =& {\int }_{S^{n-1}}{\frac{\partial (h_K{(u)}^{1-\lambda }{h}_L{(u)}^{\lambda })}{\partial \lambda }|}_{\lambda =0}d{S}_{Q_0}(u)\\ =& {\int }_{S^{n-1}}{h}_K(u)log\frac{h_L(u)}{h_K(u)}d{S}_K(u)\\ =& nV(K){\int }_{S^{n-1}}log\frac{h_L(u)}{h_K(u)}d{\overline{V}}_K(u)\\ =& nV(K)log{\overline{V}}_0(K,L).\end{array}$$

(3.7)

On the other hand, by (3.6) and (1.2), we have

$$Q_{\lambda }=(1-\lambda )\cdot K{+}_0\lambda \cdot L.$$

Therefore

$${\frac{dV(Q_{\lambda })}{d\lambda }|}_{\lambda =0}=\underset{\lambda \to 0}{lim}\frac{V((1-\lambda )\cdot K{+}_0\lambda \cdot L)-V(K)}{\lambda }.$$

(3.8)

Hence, combining (3.7) and (3.8), we obtain

$$nV(K)log{\overline{V}}_0(K,L)=\underset{\lambda \to 0}{lim}\frac{V((1-\lambda )\cdot K{+}_0\lambda \cdot L)-V(K)}{\lambda }.$$

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4 Other results and comments

Firstly, we prove that $L_0$-Minkowski combination and normalized $L_0$-mixed volume are invariant under simultaneous unimodular centro-affine transformations.

Proposition 4.1 Suppose that K and L are convex bodies in ${\mathbb{R}}^n$ that contain the origin in their interiors, and $\lambda \in [0,1]$. If $\varphi \in \mathit{SL}(n)$, then

$$\varphi ((1-\lambda )\cdot K{+}_0\lambda \cdot L)=(1-\lambda )\cdot \varphi K{+}_0\lambda \cdot \varphi L.$$

Proof For $x\in {\mathbb{R}}^n$ and $u\in S^{n-1}$, let $y=\varphi x$ and ${\varphi }^{-t}u=|{\varphi }^{-t}u|v$, then $y\in {\mathbb{R}}^n$ and $v\in S^{n-1}$. Thus we have

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Proposition 4.2 Suppose that K and L are convex bodies in ${\mathbb{R}}^n$ that contain the origin in their interiors, then for $\varphi \in \mathit{SL}(n)$ we have

$${\overline{V}}_0(\varphi K,\varphi L)={\overline{V}}_0(K,L).$$

Proof

From Theorem 1.7, we have

$${\overline{V}}_0(K,L)=exp\left\{\frac{1}{nV(K)}\underset{\lambda \to 0}{lim}\frac{V((1-\lambda )\cdot K{+}_0\lambda \cdot L)-V(K)}{\lambda }\right\}.$$

Then, by Proposition 4.1, we obtain

$$\begin{array}{rcl}{\overline{V}}_0(\varphi K,\varphi L)& =& exp\left\{\frac{1}{nV(\varphi K)}\underset{\lambda \to 0}{lim}\frac{V((1-\lambda )\cdot \varphi K{+}_0\lambda \cdot \varphi L)-V(\varphi K)}{\lambda }\right\}\\ =& exp\left\{\frac{1}{nV(\varphi K)}\underset{\lambda \to 0}{lim}\frac{V(\varphi ((1-\lambda )\cdot K{+}_0\lambda \cdot L))-V(\varphi K)}{\lambda }\right\}\\ =& exp\left\{\frac{1}{nV(K)}\underset{\lambda \to 0}{lim}\frac{V((1-\lambda )\cdot K{+}_0\lambda \cdot L)-V(K)}{\lambda }\right\}\\ =& {\overline{V}}_0(K,L).\end{array}$$

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The following proposition shows the property of weak convergence of the cone-volume probability measure.

Proposition 4.3 If $K_i$ is a sequence of convex bodies in ${\mathbb{R}}^n$ that contain the origin in their interiors, and ${lim}_{i\to \mathrm{\infty }}{K}_i=K_0$, where $K_0$ is a convex body that also contains the origin in its interior, then ${lim}_{i\to \mathrm{\infty }}{\overline{V}}_{K_i}={\overline{V}}_{K_0}$ weakly.

Proof Suppose that $f\in C(S^{n-1})$. Since $K_i\to K_0$, by definition, $h_{K_i}\to h_{K_0}$ uniformly on $S^{n-1}$. Since the continuous function $h_{K_0}$ is positive, the $h_{K_i}$ are uniformly bounded away from zero, and thus

$$f{h}_{K_i}\to f{h}_{K_0}\text{uniformly on }{S}^{n-1}.$$

But $K_i\to K_0$ also implies (see [8]) that

$$S(K_i,\cdot )\to S(K_0,\cdot )\text{weakly on }{S}^{n-1}.$$

By the continuity of the volume, that is, if $K_i\to K_0$ then $V(K_i)\to V(K_0)$, we have

$$\frac{1}{nV(K_i)}{\int }_{S^{n-1}}f(u)h_{K_i}(u)dS(K_i,u)\to \frac{1}{nV(K_0)}{\int }_{S^{n-1}}f(u)h_{K_0}(u)dS(K_0,u)$$

or, equivalently,

$${\int }_{S^{n-1}}f(u)d{\overline{V}}_{K_i}(u)\to {\int }_{S^{n-1}}f(u)d{\overline{V}}_{K_0}(u).$$

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The continuity of the normalized $L_0$-mixed volume is contained in the following proposition.

Proposition 4.4 Suppose that $K_i$ and $L_i$ are two sequences of convex bodies in ${\mathbb{R}}^n$ that contain the origin in their interiors, and ${lim}_{i\to \mathrm{\infty }}{K}_i=K$, ${lim}_{i\to \mathrm{\infty }}{L}_i=L$, where K and L are convex bodies that also contain the origin in its interior, then ${lim}_{i\to \mathrm{\infty }}{\overline{V}}_0(K_i,L_i)={\overline{V}}_0(K,L)$.

Proof Since $K_i\to K$ and $L_i\to L$, by definition, $h_{K_i}\to h_K$ and $h_{L_i}\to h_L$ uniformly on $S^{n-1}$. Since the continuous functions $h_K$ and $h_L$ are positive, the $h_{K_i}$ and $h_{L_i}$ are uniformly bounded away from zero. It follows that $\frac{h_{L_i}}{h_{K_i}}\to \frac{h_L}{h_K}$ uniformly on $S^{n-1}$, and thus that

$$log\frac{h_{L_i}}{h_{K_i}}\to log\frac{h_L}{h_K}\text{uniformly on }{S}^{n-1}.$$

By Proposition 4.3, $K_i\to K$ implies that

$${\overline{V}}_{K_i}\to {\overline{V}}_K\text{weakly on }{S}^{n-1}.$$

Hence

$$exp({\int }_{S^{n-1}}log\frac{h_{L_i}}{h_{K_i}}d{\overline{V}}_{K_i})\to exp({\int }_{S^{n-1}}log\frac{h_L}{h_K}d{\overline{V}}_K).$$

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Next, we show some properties of the Wulff shape in the following propositions.

Proposition 4.5 [8]

Suppose that ν is a Borel measure on $S^{n-1}$ and that $f_i$, f are positive continuous functions on $S^{n-1}$. If ${lim}_{i\to \mathrm{\infty }}{f}_i(u)=f(u)$ uniformly on $u\in supp\nu$, then ${lim}_{i\to \mathrm{\infty }}{W}_{\nu ,f_i}=W_{\nu ,f}$.

Proposition 4.6 Suppose that ν is a Borel measure on $S^{n-1}$ and that f, g are positive continuous functions on $S^{n-1}$, then $W_{\nu ,min\{f,g\}}=W_{\nu ,f}\cap W_{\nu ,g}$.

Proof Assume that $x\in W_{\nu ,min\{f,g\}}$, then

$$x\cdot u\le min\{f(u),g(u)\}\text{for all }u\in supp\nu .$$

Thus

$$x\cdot u\le f(u)\text{and}x\cdot u\le g(u)\text{for all }u\in supp\nu .$$

Hence, by (2.5), we get

$$x\in W_{\nu ,f}\text{and}x\in W_{\nu ,g}.$$

Then $x\in W_{\nu ,f}\cap W_{\nu ,g}$. Therefore, $W_{\nu ,min\{f,g\}}\subseteq W_{\nu ,f}\cap W_{\nu ,g}$.

Conversely, assume that $x\in W_{\nu ,f}\cap W_{\nu ,g}$, then

$$x\cdot u\le f(u)\text{and}x\cdot u\le g(u)\text{for all }u\in supp\nu .$$

Fix $u_0\in supp\nu$, then $x\cdot u_0\le min\{f(u_0),g(u_0)\}$. By the arbitrariness of $u_0$, we have $x\cdot u\le min\{f(u),g(u)\}$ for all $u\in supp\nu$. Hence $x\in W_{\nu ,min\{f,g\}}$. Therefore, $W_{\nu ,f}\cap W_{\nu ,g}\subseteq W_{\nu ,min\{f,g\}}$. □

Proposition 4.7 Suppose that ν is a Borel measure on $S^{n-1}$ and that f, g are positive continuous functions on $S^{n-1}$, then $W_{\nu ,max\{f,g\}}\supseteq W_{\nu ,f}\cup W_{\nu ,g}$.

Proof Assume that $x\in W_{\nu ,f}\cup W_{\nu ,g}$. Fix $u_0\in supp\nu$, then $x\cdot u_0\le f(u_0)$ or $x\cdot u_0\le g(u_0)$, thus $x\cdot u_0\le max\{f(u_0),g(u_0)\}$. By the arbitrariness of $u_0$, we have $x\cdot u\le max\{f(u),g(u)\}$ for all $u\in supp\nu$. Hence $x\in W_{\nu ,max\{f,g\}}$. □

Proposition 4.8 Suppose that ν is a Borel measure on $S^{n-1}$ and that f, g are positive continuous functions on $S^{n-1}$, then for real number $\lambda \in [0,1]$ we have $W_{\nu ,f^{1-\lambda }\cdot g^{\lambda }}\supseteq (1-\lambda )\cdot W_{\nu ,f}{+}_0\lambda \cdot W_{\nu ,g}$.

Proof Let $x\in (1-\lambda )\cdot W_{\nu ,f}{+}_0\lambda \cdot W_{\nu ,g}$. By (1.2), we have $x\cdot u\le h_{W_{\nu ,f}}{(u)}^{1-\lambda }{h}_{W_{\nu ,g}}{(u)}^{\lambda }$ for all $u\in supp\nu$. From (2.6), we know that $h_{W_{\nu ,f}}(u)\le f(u)$ and $h_{W_{\nu ,g}}(u)\le g(u)$ for all $u\in supp\nu$. Since f and g are positive continuous functions on $S^{n-1}$, then

$$h_{W_{\nu ,f}}{(u)}^{1-\lambda }{h}_{W_{\nu ,g}}{(u)}^{\lambda }\le f{(u)}^{1-\lambda }g{(u)}^{\lambda }.$$

Hence $x\cdot u\le f{(u)}^{1-\lambda }g{(u)}^{\lambda }$ for all $u\in supp\nu$. Therefore, by (2.5), we have $x\in W_{\nu ,f^{1-\lambda }\cdot g^{\lambda }}$. □

Proposition 4.9 Suppose that ν is a Borel measure on $S^{n-1}$ and that f, g are positive continuous functions on $S^{n-1}$, then $W_{\nu ,f+g}\supseteq W_{\nu ,f}+W_{\nu ,g}$.

Proof Let $x\in W_{\nu ,f}+W_{\nu ,g}$. Assume $x=x_1+x_2$, where $x_1\in W_{\nu ,f}$ and $x_2\in W_{\nu ,g}$, then by (2.5) we have $x_1\cdot u\le f(u)$ and $x_2\cdot u\le g(u)$ for all $u\in supp\nu$. Thus $x_1\cdot u+x_2\cdot u\le f(u)+g(u)$ for all $u\in supp\nu$. That is, $x\cdot u\le f(u)+g(u)$ for all $u\in supp\nu$. Therefore, $x\in W_{\nu ,f+g}$. □