The point vortex model for the Euler equation

1.   Introduction

The radial addition $K\widetilde{+}L$ of star sets $K$ and $L$ can be defined by

where a star set is a compact set that is star-shaped at $o$ and contains $o$ and $\rho(K, \cdot)$ denotes the radial function of star set $K$. The radial function is defined by

for $u\in S^{n-1}$, where $S^{n-1}$ denotes the surface of the unit ball centered at the origin. The initial study of the radial addition can be found in [1, p. 235]. $K$ is called a star body if $\rho(K, \cdot)$ is positive and continuous, and let ${\cal S}^{n}$ denote the set of star bodies. The radial addition and volume are the basis and core of the dual Brunn-Minkowski theory (see, e.g., ^{[2,3,4,5,6,7,8,9,10]}). It is important that the dual Brunn-Minkowski theory can count among its successes the solution of the Busemann-Petty problem in ^{[3,11,12,13,14]}. Recently, it has turned to a study extending from $L_p$-dual Brunn-Minkowski theory to Orlicz dual Brunn-Minkowski theory. The Orlicz dual Brunn-Minkowski theory and its dual have attracted people's attention ^{[15,16,17,18,19,20,21,22,23,24,25,26,27,28]}.

For $K\in {\cal S}^{n}$ and $u\in S^{n-1}$, the half chord of $K$ in the direction $u$ is defined by

If there exists a constant $\lambda > 0$ such that $d(K, u) = \lambda d(L, u),$ for all $u\in S^{n-1}$, then star bodies $K, L$ are said to have similar chord (see Gardner ^[1] or Schneider ^[29]). Lu ^[30] introduced the $i$-th chord integral of star bodies: For $K\in {\cal S}^{n}$ and $0\leq i < n$, the $i$-th chord integral of $K$, is denoted by $B_{i}(K),$ is defined by

Obviously, for $i = 0$, $B_{i}(K)$ becomes the chord integral $B(K)$.

The main aim of the present article is to generalize the chord integrals to Orlicz space. We introduce a new affine geometric quantity which we shall call Orlicz mixed chord integrals. The fundamental notions and conclusions of the chord integral and related isoperimetric inequalities for the chord integral are extended to an Orlicz setting. The new inequalities in special cases yield the $L_p$-dual Minkowski and $L_p$-dual Brunn-Minkowski inequalities for the $L_{p}$-mixed chord integrals. The related concepts and inequalities of $L_{p}$-mixed chord integrals are derived. As extensions, Orlicz multiple mixed chord integrals and Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed chord integrals are also derived.

In Section 3, we introduce the following new notion of Orlicz chord addition of star bodies.

Orlicz chord addition Let $K$ and $L$ be star bodies, the Orlicz chord addition of $K$ and $L$, is denoted by $K\check{+}_{\phi}L$, is defined by

where $u\in S^{n-1}$, and $\phi\in \Phi_{2}$, which is the set of convex functions $\phi:[0, \infty)^{2}\rightarrow (0, \infty)$ that are decreasing in each variable and satisfy $\phi(0, 0) = \infty$ and $\phi(\infty, 1) = \phi(1, \infty) = 1$.

The particular instance of interest corresponds to using (1.3) with $\phi(x_{1}, x_{2}) = \phi_{1}(x_{1})+\varepsilon\phi_{2}(x_{2})$ for $\varepsilon > 0$ and some $\phi_{1}, \phi_{2}\in\Phi$, which are the sets of convex functions $\phi_{1}, \phi_{2}:[0, \infty)\rightarrow (0, \infty)$ that are decreasing and satisfy $\phi_{1}(0) = \phi_{2}(0) = \infty$, $\phi_{1}(\infty) = \phi_{2}(\infty) = 0$ and $\phi_{1}(1) = \phi_{2}(1) = 1.$

In accordance with the spirit of Aleksandrov ^[31], Fenchel and Jessen's ^[32] introduction of mixed quermassintegrals, and introduction of Lutwak's ^[33] $L_p$-mixed quermassintegrals, we are based on the study of first-order variations of the chord integrals. In Section 4, we prove that the first order Orlicz variation of the mixed chord integral can be expressed as: For $K, L\in {\cal S}^{n},$ $\phi_{1}, \phi_{2}\in {\Phi}$, $0\leq i < n$ and $\varepsilon > 0$,

where $(\phi_{1})'_{r}(1)$ denotes the value of the right derivative of convex function $\phi_{1}$ at point $1$. In this first order variational equation (1.4), we find a new geometric quantity. Based on this, we extract the required geometric quantity, denoted by $B_{\phi, i}(K, L)$ which we shall call Orlicz mixed chord integrals of $K$ and $L$, as follows

We show also that the new affine geometric quantity has an integral representation as follows:

When $\phi(t) = t^{-p}$ and $p\geq 1$, the new affine geometric quantity becomes a new $L_{p}$-mixed chord integrals of $K$ and $L$, denoted by $B_{p, i}(K, L)$, which as is in (2.7).

In Section 5, we establish an Orlicz Minkowski inequality for the mixed chord and Orlicz mixed chord integrals.

Orlicz Minkowski inequality for the Orlicz mixed chord integrals If $K, L\in {\cal S}^{n}$, $0\leq i < n$ and $\phi\in \Phi$, then

If $\phi$ is strictly convex, the equality holds if and only if $K$ and $L$ are similar chord.

When $\phi(t) = t^{-p}$ and $p\geq 1$, (1.7) becomes a new $L_{p}$-Minkowski inequality (2.8) for the $L_{p}$-mixed chord integrals.

In Section 6, as an application, we establish an Orlicz Brunn-Minkowski inequality for the Orlicz chord additions and the mixed chord integrals:

Orlicz Brunn-Minkowski inequality for the Orlicz chord additions If $K, L\in{\cal S}^{n}$, $0\leq i < n$ and $\phi\in \Phi_{2}$, then

If $\phi$ is strictly convex, the equality holds if and only if $K$ and $L$ are similar chord.

When $\phi(t) = t^{-p}$ and $p\geq 1$, (1.8) becomes a new $L_{p}$-Brunn-Minkowski inequality (2.9) for the mixed chord integrals.

A new isoperimetric inequality for the chord integrals is given in Section 7. In Section 8, Orlicz multiple mixed chord integrals is introduced and Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed chord integrals is established.

2.   Preliminaries

The setting for this paper is $n$-dimensional Euclidean space ${\Bbb R}^{n}$. A body in ${\Bbb R}^{n}$ is a compact set equal to the closure of its interior. For a compact set $K\subset {\Bbb R}^{n}$, we write $V(K)$ for the ($n$-dimensional) Lebesgue measure of $K$ and call this the volume of $K$. Associated with a compact subset $K$ of ${\Bbb R}^n$ which is star-shaped with respect to the origin and contains the origin, its radial function is $\rho(K, \cdot): S^{n-1}\rightarrow [0, \infty)$ is defined by

Note that the class (star sets) is closed under union, intersection, and intersection with subspace. The radial function is homogeneous of degree $-1$, that is (see e.g. ^[1]),

for all $u\in {S}^{n-1}$ and $r > 0$. Let $\tilde{\delta}$ denote the radial Hausdorff metric, as follows: if $K, L\in {\cal S}^{n}$, then

From the definition of the radial function, it follows immediately that for $A\in GL(n)$ the radial function of the image $AK = \{Ay: y\in K\}$ of $K$ is given by (see e.g. ^[29])

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

For $K_{i}\in {\cal S}^{n}, i = 1, \ldots, m$, define the real numbers $R_{K_{i}}$ and $r_{K_{i}}$ by

Obviously, $0 < r_{K_{i}} < R_{K_{i}},$ for all $K_{i}\in {\cal S}^{n}$. Writing $R = \max\{R_{K_{i}}\}$ and $r = \min\{r_{K_{i}}\}$, where $i = 1, \ldots, m.$

2.1.   Mixed chord integrals

If $K_{1}, \ldots, K_{n}\in {\cal S}^{n}$, the mixed chord integral of $K_{1}, \ldots, K_{n}$, is denoted by $B(K_{1}, \ldots, K_{n})$, is defined by (see ^[30])

If $K_{1} = \cdots = K_{n-i} = K,$ $K_{n-i+1} = \cdots = K_{n} = L$, the mixed chord integral $B(K_{1}, \ldots, K_{n})$ is written as $B_{i}(K, L)$. If $L = B$ ($B$ is the unit ball centered at the origin), the mixed chord integral $B_{i}(K, L) = B_{i}(K, B)$ is written as $B_{i}(K)$ and called the $i$-th chord integral of $K$. Obviously, For $K\in {\cal S}^{n}$ and $0\leq i < n$, we have

If $K_{1} = \cdots = K_{n-i-1} = K,$ $K_{n-i} = \cdots = K_{n-1} = B$ and $K_{n} = L$, the mixed chord integral $B(\underbrace{K, \ldots, K}_{n-i-1}, \underbrace{B, \ldots, B}_{i}, L)$ is written as $B_{i}(K, L)$ and called the $i$-th mixed chord integral of $K$ and $L$. For $K, L\in {\cal S}^{n}$ and $0\leq i < n$, it is easy to see that

This integral representation (2.4), together with the Hölder inequality, immediately give the Minkowski inequality for the $i$-th mixed chord integral: If $K, L\in {\cal S}^{n}$ and $0\leq i < n$, then

with equality if and only if $K$ and $L$ are similar chord.

2.2.   $L_p$-mixed chord integrals

Definition 2.1 (The $L_p$-chord addition) Let $K, L\in {\cal S}^{n}$ and $p\geq 1$, the $L_{p}$ chord addition $\check{+}_{p}$ of star bodies $K$ and $L$, is defined by

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

Obviously, putting $\phi(x_{1}, x_{2}) = x_{1}^{-p}+x_{2}^{-p}$ and $p\geq 1$ in (1.3), (1.3) becomes (2.6). The following result follows immediately from (2.6) with $p\geq 1$.

Definition 2.2 (The $L_p$-mixed chord integrals) Let $K, L\in {\cal S}^{n}$, $0\leq i < n$ and $p\geq 1$, the $L_p$-mixed chord integral of star $K$ and $L$, denoted by $B_{p, i}(K, L)$, is defined by

Obviously, when $K = L$, the $L_p$-mixed chord integral $B_{p, i}(K, K)$ becomes the $i$-th chord integral $B_{i}(K).$ This integral representation (2.7), together with the Hölder inequality, immediately gives:

Proposition 2.3 If $K, L\in {\cal S}^{n}$, $0\leq i < n$ and $p\geq 1$, then

with equality if and only if $K$ and $L$ are similar chord.

Proposition 2.4 If $K, L\in {\cal S}^{n}$, $0\leq i < n$ and $p\geq 1$, then

with equality if and only if $K$ and $L$ are similar chord.

Proof From (2.6) and (2.7), it is easily seen that the $L_p$-chord integrals is linear with respect to the $L_p$-chord addition, and together with inequality (2.8), we have for $p\geq 1$

with equality if and only if $K$ and $L$ are similar chord.

Take $K\check{+}_{p}L$ for $Q$, recall that $B_{p, i}(Q, Q) = B_{i}(Q)$, inequality (2.9) follows easily.

3.   Orlicz chord addition

Throughout this paper, the standard orthonormal basis for ${\Bbb R}^{n}$ will be $\{e_{1}, \ldots, e_{n}\}$. Let $\Phi_{n},$ $n\in{\Bbb N}$, denote the set of convex functions $\phi:[0, \infty)^{n} \rightarrow (0, \infty)$ that are strictly decreasing in each variable and satisfy $\phi(0) = \infty$ and $\phi(e_{j}) = 1$, $j = 1, \ldots, n$. When $n = 1$, we shall write $\Phi$ instead of $\Phi_{1}$. The left derivative and right derivative of a real-valued function $f$ are denoted by $(f)'_{l}$ and $(f)'_{r}$, respectively. We first define the Orlicz chord addition.

Definition 3.1 (The Orlicz chord addition) Let $m\geq 2, \phi\in\Phi_{m}$, $K_{j}\in {\cal S}^{n}$ and $j = 1, \ldots, m$, the Orlicz chord addition of $K_{1}, \ldots, K_{m}$, is denoted by $\check{+}_{\phi}(K_{1}, \ldots, K_{m})$, is defined by

for $u\in S^{n-1}.$ Equivalently, the Orlicz chord addition $\check{+}_{\phi}(K_{1}, \ldots, K_{m})$ can be defined implicitly by

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

An important special case is obtained when

for some fixed $\phi_{j}\in \Phi$ such that $\phi_{1}(1) = \cdots = \phi_{m}(1) = 1$. We then write $\check{+}_{\phi}(K_{1}, \ldots, K_{m}) = K_{1}\check{+}_{\phi}\cdots\check{+}_{\phi}K_{m}.$ This means that $K_{1}\check{+}_{\phi}\cdots\check{+}_{\phi}K_{m}$ is defined either by

for all $u\in {S}^{n-1}$, or by the corresponding special case of (3.2).

Lemma 3.2 The Orlicz chord addition $\check{+}_{\phi}: ({\cal S}^{n})^{m}\rightarrow {\cal S}^{n}$ is monotonic.

Proof This follows immediately from (3.1).

Lemma 3.3 The Orlicz chord addition $\check{+}_{\phi}: ({\cal S}^{n})^{m}\rightarrow {\cal S}^{n}$ is $GL(n)$ covariant.

Proof From (2.1), (3.1) and let $A\in GL(n)$, we obtain

This shows Orlicz chord addition $\check{+}_{\phi}$ is $GL(n)$ covariant.

Lemma 3.4 Suppose $K_{1}, \ldots, K_{m}\in{\cal S}^{n}$. If $\phi\in \Phi$, then

if and only if

Proof This follows immediately from Definition 3.1.

Lemma 3.5 Suppose $K_{m}, \ldots, K_{m}\in{\cal S}^{n}$. If $\phi\in \Phi$, then

Proof Suppose $d(\check{+}_{\phi}(K_{1}, \ldots, K_{m}), u) = t$, from Lemma 3.4 and noting that $\phi$ is strictly deceasing on $(0, \infty)$, we have

Noting that the inverse $\phi^{-1}$ is strictly deceasing on $(0, \infty)$, we obtain the lower bound for $d(\check{+}_{\phi}(K_{1}, \ldots, K_{m}), u)$:

To obtain the upper estimate, observe the fact from the Lemma 3.4, together with the convexity and the fact $\phi$ is strictly deceasing on $(0, \infty)$, we have

Then we obtain the upper estimate:

Lemma 3.6 The Orlicz chord addition $\check{+}_{\phi}: ({\cal S}^{n})^{m}\rightarrow {\cal S}^{n}$ is continuous.

Proof To see this, indeed, let $K_{ij}\in {\cal S}^{n},$ $i \in {\Bbb N}\cup\{0\},$ $j = 1, \ldots, m,$ be such that $K_{ij}\rightarrow K_{0j}$ as $i\rightarrow\infty$. Let

Then Lemma 3.5 shows

where $r_{ij} = \min\{r_{K_{ij}}\}$ and $R_{ij} = \max\{R_{K_{ij}}\}.$ Since $K_{ij}\rightarrow K_{0j}$, we have $R_{K_{ij}}\rightarrow R_{K_{0j}} < \infty$ and $r_{K_{ij}}\rightarrow r_{K_{0j}} > 0$, and thus there exist $a, b$ such that $0 < a\leq t_{i}\leq b < \infty$ for all $i$. To show that the bounded sequence $\{t_{i}\}$ converges to $d(\check{+}_{\phi}(K_{01}, \ldots, K_{0m}), u),$ we show that every convergent subsequence of $\{t_{i}\}$ converges to $d(\check{+}_{\phi}(K_{01}, \ldots, K_{0m}), u)$. Denote any subsequence of $\{t_{i}\}$ by $\{t_{i}\}$ as well, and suppose that for this subsequence, we have

Obviously $a\leq t_{*}\leq b.$ Noting that $\phi$ is a continuous function, we obtain

Hence

as $i\rightarrow\infty$.

This shows that the Orlicz chord addition $\check{+}_{\phi}: ({\cal S}^{n})^{m}\rightarrow {\cal S}^{n}$ is continuous.

Next, we define the Orlicz chord linear combination for the case $m = 2$.

Definition 3.7 (The Orlicz chord linear combination) The Orlicz chord linear combination, denoted by $\check{+}_{\phi}(K, L, \alpha, \beta)$ for $K, L\in {\cal S}^{n}$, and $\alpha, \beta\geq 0$ (not both zero), is defined by

for $\phi_{1}, \phi_{2}\in \Phi$ and all $u\in {S}^{n-1}$.

We shall write $K\check{+}_{\phi}\varepsilon\cdot L$ instead of $\check{+}_{\phi}(K, L, 1, \varepsilon)$, for $\varepsilon\geq 0$ and assume throughout that this is defined by (3.1), if $\alpha = 1, \beta = \varepsilon$ and $\phi\in \Phi$. We shall write $K\check{+}_{\phi}L$ instead of $\check{+}_{\phi}(K, L, 1, 1)$ and call it the Orlicz chord addition of $K$ and $L$.

4.   Orlicz mixed chord integrals

In order to define Orlicz mixed chord integrals, we need the following Lemmas 4.1-4.4.

Lemma 4.1 Let $\phi\in \Phi$ and $\varepsilon > 0$. If $K, L\in {\cal S}^{n}$, then $K\check{+}_{\phi}\varepsilon\cdot L\in {\cal S}^{n}.$

Proof Let $u_{0}\in S^{n-1}$, and $\{u_{i}\}\subset S^{n-1}$ be any subsequence such that $u_{i}\rightarrow u_{0}$ as $i\rightarrow \infty.$

Let

Then Lemma 3.5 shows

where $R = \max\{R_{K}, R_{L}\}$ and $r = \min\{r_{K}, r_{L}\}.$

Since $K, L\in {\cal S}^{n}$, we have $0 < r_{K}\leq R_{K} < \infty$ and $0 < r_{L}\leq R_{L} < \infty$, and thus there exist $a, b$ such that $0 < a\leq \lambda_{i}\leq b < \infty$ for all $i$. To show that the bounded sequence $\{\lambda_{i}\}$ converges to $d(K\check{+}_{\phi}\varepsilon\cdot L, u_{0}),$ we show that every convergent subsequence of $\{\lambda_{i}\}$ converges to $d(K\check{+}_{\phi}\varepsilon\cdot L, u_{0})$. Denote any subsequence of $\{\lambda_{i}\}$ by $\{\lambda_{i}\}$ as well, and suppose that for this subsequence, we have

Obviously $a\leq \lambda_{0}\leq b.$ From (3.4) and note that $\phi_{1}, \phi_{2}$ are continuous functions, so $\phi^{-1}_{1}$ is continuous, we obtain

as $i\rightarrow \infty.$ Hence

Therefore

That is

as $i\rightarrow \infty.$

This shows that $K\check{+}_{\phi}\varepsilon\cdot L\in {\cal S}^{n}.$

Lemma 4.2 If $K, L\in {\cal S}^{n}$, $\varepsilon > 0$ and $\phi\in \Phi$, then

as $\varepsilon\rightarrow 0^{+}$.

Proof This follows immediately from (3.4).

Lemma 4.3 If $K, L\in {\cal S}^{n}$, $0\leq i < n$ and $\phi_{1}, \phi_{2}\in \Phi$, then

Proof From (3.4), Lemma 4.2 and notice that $\phi_{1}^{-1}$, $\phi_{2}$ are continuous functions, we obtain for $0\leq i < n$

where

and note that $y\rightarrow 1^{-}$ as $\varepsilon\rightarrow 0^{+}.$

Lemma 4.4 If $\phi\in \Phi_{2}$, $0\leq i < n$ and $K, L\in {\cal S}^{n}$, then

Proof This follows immediately from (2.1) and Lemma 4.3.

Denoting by $B_{\phi, i}(K, L)$, for any $\phi\in\Phi$ and $0\leq i < n$, the integral on the right-hand side of (4.3) with $\phi_{2}$ replaced by $\phi$, we see that either side of the equation (4.3) is equal to $B_{\phi_{2}, i}(K, L)$ and hence this new Orlicz mixed chord integrals $B_{\phi, i}(K, L)$ has been born.

Definition 4.5 (The Orlicz mixed chord integral) For $\phi\in \Phi$ and $0\leq i < n$, Orlicz mixed chord integral of star bodies $K$ and $L$, $B_{\phi, i}(K, L)$, is defined by

Lemma 4.6 If $\phi_{1}, \phi_{2}\in \Phi$, $0\leq i < n$ and $K, L\in {\cal S}^{n}$, then

Proof This follows immediately from Lemma 4.4 and (4.4).

Lemma 4.7 If $K, L\in {\cal S}^{n}$, $\phi\in\Phi$ and any $A\in SL(n)$, then for $\varepsilon > 0$

Proof This follows immediately from (2.1) and (3.3).

We find easily that $B_{\phi, i}(K, L)$ is invariant under simultaneous unimodular centro-affine transformation.

Lemma 4.8 If $\phi\in \Phi$, $0\leq i < n$ and $K, L\in {\cal S}^{n}$, then for $A\in SL(n)$,

Proof This follows immediately from Lemmas 4.6 and 4.7.

5.   Orlicz chord Minkowski inequality

In this section, we will define a Borel measure in $S^{n-1}$, denoted by $B_{n, i}(K, \upsilon),$ which we shall call the chord measure of star body $K$.

Definition 5.1 (The chord measure) Let $K\in {\cal S}^{n}$ and $0\leq i < n$, the chord measure of star body $K$, denoted by $B_{n, i}(K, \upsilon),$ is defined by

Lemma 5.2 (Jensen's inequality) Let $\mu$ be a probability measure on a space $X$ and $g: X\rightarrow I\subset {\Bbb R}$ be a $\mu$-integrable function, where $I$ is a possibly infinite interval. If $\psi: I\rightarrow {\Bbb R}$ is a convex function, then

If $\psi$ is strictly convex, the equality holds if and only if $g(x)$ is constant for $\mu$-almost all $x\in X$ (see [34, p. 165]).

Lemma 5.3 Suppose that $\phi: [0, \infty)\rightarrow (0, \infty)$ is decreasing and convex with $\phi(0) = \infty$. If $K, L\in{\cal S}^{n}$ and $0\leq i < n$, then

If $\phi$ is strictly convex, the equality holds if and only if $K$ and $L$ are similar chord.

Proof For $K\in {\cal S}^{n-1}$, $0\leq i < n$ and any $u\in S^{n-1}$, the chord measure $\frac{ d(K, u)^{n-i}}{ n{B}_{i}(K)}dS(u)$ is a probability measure on $S^{n-1}$. Hence, from (2.4), (2.5), (5.1) and by using Jensen's inequality, and in view of $\phi$ is decreasing, we obtain

Next, we discuss the equality in (5.3). If $\phi$ is strictly convex, suppose the equality holds in (5.3), form the equality necessary conditions of Jensen's inequality and (2.5), it follows that $d(L, u)/d(K, u)$ is constant, and $K$ and $L$ are similar chord, respectively. This yields that there exists $r > 0$ such that $d(L, u) = rd(K, u),$ for all $u\in S^{n-1}$. On the other hand, suppose that $K$ and $L$ are similar chord, i.e. there exists $\lambda > 0$ such that $d(L, u) = \lambda d(K, u)$ for all $u\in S^{n-1}$. Hence

This implies the equality in (5.3) holds.

Theorem 5.4 (Orlicz chord Minkowski inequality) If $K, L\in {\cal S}^{n}$, $0\leq i < n$ and $\phi\in \Phi$, then

If $\phi$ is strictly convex, the equality holds if and only if $K$ and $L$ are similar chord.

Proof This follows immediately from (4.4) and Lemma 5.3.

Corollary 5.5 If $K, L\in {\cal S}^{n}$, $0\leq i < n$ and $p\geq 1$, then

with equality if and only if $K$ and $L$ are similar chord.

Proof This follows immediately from Theorem 5.4 with $\phi_{1}(t) = \phi_{2}(t) = t^{-p}$ and $p\geq 1$.

Taking $i = 0$ in (5.5), this yields $L_{p}$-Minkowski inequality: If $K, L\in {\cal S}^{n}$ and $p\geq 1$, then

with equality if and only if $K$ and $L$ are similar chord.

Corollary 5.6 Let $K, L\in {\cal M}\subset{\cal S}^{n}$, $0\leq i < n$ and $\phi\in \Phi$, and if either

or

then $K = L.$

Proof Suppose (5.6) holds. Taking $K$ for $Q$, then from (2.3), (4.4) and (5.3), we obtain

with equality if and only if $K$ and $L$ are similar chord. Hence

with equality if and only if $K$ and $L$ are similar chord. On the other hand, if taking $L$ for $Q$, by similar arguments, we get $B_{i}(K)\geq B_{i}(L),$ with equality if and only if $K$ and $L$ are similar chord. Hence $B_{i}(K) = B_{i}(L),$ and $K$ and $L$ are similar chord, it follows that $K$ and $L$ must be equal.

Suppose (5.7) holds. Taking $L$ for $Q$, then from from (2.3), (4.4) and (5.3), we obtain

with equality if and only if $K$ and $L$ are similar chord. Hence

with equality if and only if $K$ and $L$ are similar chord. On the other hand, if taking $K$ for $Q$, by similar arguments, we get $B_{i}(K)\geq B_{i}(L),$ with equality if and only if $K$ and $L$ are similar chord. Hence $B_{i}(K) = B_{i}(L),$ and $K$ and $L$ have similar chord, it follows that $K$ and $L$ must be equal.

When $\phi_{1}(t) = \phi_{2}(t) = t^{-p}$ and $p\geq 1$, Corollary 5.6 becomes the following result.

Corollary 5.7 Let $K, L\in {\cal M}\subset{\cal S}^{n}$, $0\leq i < n$ and $p\geq 1$, and if either

or

then $K = L.$

6.   Orlicz chord Brunn-Minkowski inequality

Lemma 6.1 If $K, L\in {\cal S}^{n}$, $0\leq i < n$, and $\phi_{1}, \phi_{2}\in \Phi$, then

Proof From (3.1), (3.4) and (4.4), we have for $K\check{+}_{\phi}L = Q\in {\cal S}^{n}$

The completes the proof.

Lemma 6.2 Let $K, L\in {\cal S}^{n}$, $\varepsilon > 0$ and $\phi\in\Phi$.

(1) If $K$ and $L$ are similar chord, then $K$ and $K\check{+}_{\phi}\varepsilon\cdot L$ are similar chord.

(2) If $K$ and $K\check{+}_{\phi}\varepsilon\cdot L$ are similar chord, then $K$ and $L$ are similar chord.

Proof Suppose exist a constant $\lambda > 0$ such that $d(L, u) = \lambda d(K, u)$, we have

On the other hand, the exist unique constant $\delta > 0$ such that

where $\delta$ satisfies that

This shows that $d(K\check{+}_{\phi}\varepsilon\cdot L, u) = \delta d(K, u).$

Suppose exist a constant $\lambda > 0$ such that $d(K\check{+}_{\phi}\varepsilon\cdot L, u) = \lambda d(K, u).$ Then

This shows that

is a constant. This yields that $K\check{+}_{\phi}\varepsilon\cdot L$ and $L$ are similar chord. Namely $K$ and $L$ are similar chord.

Theorem 6.3 (Orlicz chord Brunn-Minkowski inequality) If $K, L\in{\cal S}^{n}$, $0\leq i < n$ and $\phi\in \Phi_{2}$, then

If $\phi$ is strictly convex, the equality holds if and only if $K$ and $L$ are similar chord.

Proof From (5.4) and Lemma 6.1, we have

This is just inequality (6.3). From the equality condition of (5.4) and Lemma 6.3, it yields that if $\phi$ is strictly convex, equality in (6.3) holds if and only if $K$ and $L$ are similar chord.

Corollary 6.4 If $K, L\in {\cal S}^{n}$, $0\leq i < n$ and $p\geq 1$, then

with equality if and only if $K$ and $L$ are similar chord.

Proof This follows immediately from Theorem 6.2 with $\phi(x_{1}, x_{2}) = x_{1}^{-p}+x_{2}^{-p}$ and $p\geq 1$.

Taking $i = 0$ in (6.4), this yields the $L_{p}$-Brunn-Minkowski inequality for the chord integrals. If $K, L\in{\cal S}^{n}$ and $p\geq 1$, then

with equality if and only if $K$ and $L$ are similar chord.

7.   The isoperimetric inequality for chord integrals

As a application, in the section, we give a new isoperimetric inequality for chord integrals. As we all know, the isoperimetric inequality for convex bodies can be stated below (see e.g. ^[26], p. 318).

The isoperimetric inequality If $K$ is convex body in ${\Bbb R}^{n}$, then

with equality if and only if $K$ is an $n$-ball.

Here $B$ is the unit ball centered at the origin, $V(K)$ denotes the volume of $K$ and $S(K)$ is the surface area of $K$, defined by (see ^[26], p. 318)

where $+$ the usual Minkowski sum. Here, the mixed volume of convex bodies $K$ and $L$, $V_{1}(K, L)$, defined by (see e.g. ^[1])

Next, we give some new isoperimetric inequalities for mixed chord integrals by using the Orlicz chord Minkowski inequality established in Section 5.

Theorem 7.1 (The $L_{p}$-isoperimetric inequality for mixed chord integrals) If $K\in {\cal S}^{n}$, $0\leq i < n$ and $p\geq 1$, then

with equality if and only if $K$ is an $n$-ball, where $\widetilde{B}_{p, i}(K) = nB_{p, i}(K, B)$.

Proof Putting $L = B$, $\phi(t) = t^{-p}$ and $p\geq 1$ in Orlicz chord Minkowski inequality (5.4)

That is

Hence

From the equality of (5.4), we find that the equality in (7.3) holds if and only if $K$ and $B$ are similar chord. This yields that the equality in (7.3) holds if and only if $K$ is an $n$-ball.

Theorem 7.2 (The isoperimetric inequality for the chord integrals) If $K\in {\cal S}^{n}$, then

with equality if and only if $K$ is an $n$-ball, where $\widetilde{B}(K) = nB_{1}(K, B)$.

Proof This follows immediately from (7.3) with $p = 1$ and $i = 0$.

This is just a similar form of the classical isoperimetric inequality (7.1).

8.   Extensions

As extensions, in the Section, the Orlicz mixed chord integral of $K$ and $L$, $B_{\phi}(K, L)$, is generalized into Orlicz multiple mixed chord integral of $(n+1)$ star bodies $L_{1}, K_{1}, \ldots, K_{n}$. Further, we generalize the Orlicz-Minklowski inequality into Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed chord integrals.

Theorem 8.1 If $L_{1}, K_{1}, \ldots, K_{n}\in {\cal S}^{n}$ and ${\phi_{1}, \phi_{2}}\in\Phi$, then

Proof This may yield by using a generalized idea and method of proving Lemma 4.4. Here, we omit the details.

Obviously, (4.3) is a special case of (8.1). Moreover, from Theorem 8.1, we can find the following definition:

Definition 8.2 (Orlicz multiple mixed chord integrals) Let $L_{1}, K_{1}, \ldots, K_{n}\in {\cal S}^{n}$ and ${\phi}\in\Phi$, the Orlicz multiple mixed chord integral of $(n+1)$ star bodies $L_{1}, K_{1}, \ldots, K_{n}$, is denoted by $B_{\phi}(L_{1}, K_{1}, \ldots, K_{n})$, is defined by

When $L_{1} = K_{1}$, $B_{\phi}(L_{1}, K_{1}, \ldots, K_{n})$ becomes the well-known mixed chord integral $B(K_{1}, \ldots, K_{n})$. Obviously, for $0\leq i < n$, $B_{\phi, i}(K, L)$ is also a special case of $B_{\phi}(L_{1}, K_{1}, \ldots, K_{n})$.

Corollary 8.3 If $L_{1}, K_{1}, \ldots, K_{n}\in{\cal S}^{n}$ and ${\phi_{1}, \phi_{2}}\in\Phi$, then

Proof This yields immediately from (8.1) and (8.2).

Similar to the proof of Theorem 5.4, we may establish an Orlicz-Aleksandrov-Fenchel inequality as follows:

Theorem 8.4 (Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed chord integrals) If $L_{1}, K_{1}, \ldots, K_{n}\in{\cal S}^{n}$, $\phi\in\Phi$ and $1\leq r\leq n$, then

If $\phi$ is strictly convex, equality holds if and only if $L_{1}, K_{1}, \ldots, K_{r}$ are all of similar chord.

Proof This yields immediately by using a generalized idea and method of proving Theorem 5.4. Here, we omit the details.

Obviously, the Orlicz-Minklowski inequality (5.4) is a special case of the Orlicz-Aleksandrov-Fenchel inequality (8.4). Moreover, when $L_{1} = K_{1}$, (8.4) becomes the following Aleksandrov-Fenchel inequality for the mixed chords.

Corollary 8.5 (Aleksandrov-Fenchel inequality for the mixed chord integrals) If $K_{1}, \ldots, K_{n}\in{\cal S}^{n}$ and $1\leq r\leq n$, then

with equality if and only if $K_{1}, \ldots, K_{r}$ are all of similar chord.

Finally, it is worth mentioning: when $\phi(t) = t^{-p}$ and $p\geq 1$, $B_{\phi}(L_{1}, K_{1}, \ldots, K_{n})$ written as $B_{p}(L_{1}, K_{1}, \ldots, K_{n})$ and call it $L_{p}$-multiple mixed chord integrals of $(n+1)$ star bodies $L_{1}, K_{1}, \ldots, K_{n}$. So, the new concept of $L_{p}$-multiple mixed chord integrals and $L_{p}$-Aleksandrov-Fenchel inequality for the $L_{p}$-multiple mixed chord integrals may be also derived. Here, we omit the details of all derivations.

Acknowledgments

Research is supported by National Natural Science Foundation of China (11371334, 10971205).

Conflict of interest

The author declares that no conflicts of interest in this paper.