Let $\mathbb{R}^{n}$ be the $n$-dimensional Euclidean space. A convex body $K\subseteq \mathbb{R}^{n}$ is a compact convex subset with non-empty interior. Associated with a compact convex set $K$ is its support function $h_{K}$ defined on $\mathbb{R}^{n}$ by $h_{K}(x)=\max \{\langle x, ~y\rangle: y\in K\}, $ where $\langle x, ~y\rangle$ is the usual inner product of $x$ and $y$ in $\mathbb{R}^{n}.$ The support function $h_{K}$ is positively homogeneous of degree 1. We shall usually be concerned with the restriction of the support function to the unit sphere $S^{n-1}$ in $\mathbb{R}^{n}.$

For a compact set $L$ in $\mathbb{R}^{n}$ which is star shaped with respect to the origin $o, $ define the radial function $\rho_{L}$ of $L$ by $\rho_{L}(x)= \max \{\lambda \geq 0: \lambda x \in L\}, ~x \in \mathbb{R}^{n}-\{ o \}.$ The radial function is positively homogeneous of degree $-1.$ One can identify the radial function with its restriction to the unit sphere $S^{n-1}$ in $\mathbb{R}^{n}$.

If $\rho_{L}$ is continuous, we shall call $L$ a star body. A star body which is centrally symmetric with respect to the origin will be called a centered body. We shall use $\mathscr{S}$ and $\mathscr{S}_{e}$ to denote the set of star bodies and the set of centered bodies, respectively.

For a convex body $K$ containing the origin in its interior, the polar body $K^{\ast}$ of $K$ is defined by

$ K^{\ast}= \{x\in \mathbb{R}^{n}: \langle x, ~y\rangle \leq 1, y \in K\}. $

It is easy to verify that $K^{\ast \ast}=K, $ and that $h_{K^{\ast}}=\rho^{-1}_{K}, \rho_{K^{\ast}}=h^{-1}_{K}.$

If $K$ is a centered convex body, then the reciprocal of its radial function induces a norm on $\mathbb{R}^{n}, $ denoted by $\|\centerdot\|_{K}, $ whose unit ball is $K.$ That is,

$ \|x\|_{K}=\rho_{K}(x)^{-1}, x\in \mathbb{R}^{n}. $

Conversely, if $(\mathbb{R}^{n}, ~ \|\centerdot\| )$ is a normed space with unit ball $K$ (i.e., $K=\{x\in \mathbb{R}^{n}:\|x\| \leq 1\}$), it is easily seen that $\|\centerdot\|=\|\centerdot\|_{K}.$

A centered convex body $K$ in $\mathbb{R}^{n}$ is called an $L_{p}$-ball, if it is the unit ball of an $n$-dimensional subspace of some $L_{p}$-space. Denote by $\mathcal{L}_{p}$ the class of $L_{p}$-balls. It is noted that the class $\mathcal{L}_{p}$ is well known and is important in the local theory of Banach spaces. See, for example, [11, 12].

Note that $\mathcal{L}_{2}$ is the class of centered ellipsoids. The most important example of an $L_{p}$-ball is the unit ball of space $l^{n}_{p}$ given by

$ \{x=(x_{1}, \cdots, x_{n})\in \mathbb{R}^{n}: \mid x_{1}\mid ^{p}+\cdots+\mid x_{n}\mid ^{p}\leq 1\}. $

Recall that the $L_{p}$-centroid body $\Gamma_{p}L$ of a star body $L$ is defined by

$ h_{\Gamma_{p}L}(u)^{p}=\frac{1}{V(L)}\int_{L}\mid \langle u, x\rangle \mid^{p}\ dx, u\in S^{n-1}, p\geq 1. $

Therefore the polar body $\Gamma^{\ast}_{p}L$ of $\Gamma_{p}L$ belongs to $\mathcal{L}_{p}$. The first-order centroid body $\Gamma_{1}L$ is usually called the centroid body of $L$ [23] and is denoted by $\Gamma L.$ For $p=2, $ they are the Legendre ellipsoids which appears in classical mechanics. In the important paper [16], the $L_{p}$ analogs of two famous affine isoperimetric inequalities in convex geometry, which are the Busemann-Petty centroid inequality and Petty projection inequality, are established.

Using polar coordinates, one can write the last integral as an integral over $S^{n-1}, $

$ h_{\Gamma_{p}L}(u)^{p}=\frac{1}{(n+p)V(L)}\int_{S^{n-1}}\mid \langle u, v\rangle \mid^{p}\rho_{L}(v)^{n+p}dv. $

In this paper, we establish several new characterizations and inequalities for $L_{p}$-balls by using dual mixed volumes, spherical Radon transform and Fourier transform. One of the inequalities is related to the famous maximal slice conjecture.

To make the paper self-contained, we will recall some basic facts on dual mixed volume and Radon transform. For more details, one can refer [2, 4, 9, 13-16, 25, 26], etc.

As usual, $S^{n-1}$ denotes the unit sphere, $B_{n}$ the unit ball and $o$ the origin in the $n$-dimensional Euclidean space $\mathbb{R}^{n}.$ The surface area of the unit sphere $S^{n-1}$ and the volume of the unit ball $B_{n}$ in $\mathbb{R}^{n}$ are denoted by $\alpha_{n-1}$ and $\omega_{n}$, respectively. Note that $\alpha_{n-1}=n \omega_{n}$, and $\omega_{n}=\frac{2\pi^{\frac{n}{2}}}{n\Gamma(\frac{n}{2})}.$ For real $p\geq 1, $ define $c_{n, p}$ by $c_{n, p}=\frac{\omega_{n+p}}{\omega_{2}\omega_{n}\omega_{p-1}}.$

For $K_{1}, \cdots, K_{r}\in \mathscr{S}, $ $\lambda_{1}, \cdots, \lambda_{r}\geq 0$, the radial linear combination $\lambda _{1}K_{1}\widetilde{+}\cdots \widetilde{+}\lambda _{r}K_{r} \in \mathscr{S}, $ is defined by

$ \rho_{\lambda _{1}K_{1}\widetilde{+}\cdots \widetilde{+}\lambda _{r}K_{r}}=\lambda _{1}\rho_{K_{1}} + \cdots+\lambda _{r}\rho_{K_{r}}. $

The volume of the radial linear combination $\lambda _{1}K_{1}\widetilde{+} \cdots \widetilde{+}\lambda _{r}K_{r} $ is a homogeneous $n$th-degree polynomial in the $\lambda_i, $

$ V(\lambda_{1}K_{1}\widetilde{+}\cdots \widetilde{+}\lambda _{r}K_{r})=\sum \widetilde{V}(K_{i_1}, \cdots, K_{i_n})\lambda_{i_1}\cdots\lambda_{i_n}, $

where the sum is taken over all $n$-tuples $(i_{1}, \cdots, i_{n})$ whose entries are positive integers not exceeding $r$. The coefficient $\widetilde{V}(K_{i_1}, \cdots, K_{i_n})$ is non-negative and depends only on the bodies $K_{i_1}, \cdots, K_{i_n}.$ It is called the dual mixed volume of $K_{i_1}, \cdots, K_{i_n}.$ One has the following integral representation of dual mixed volumes:

$ \widetilde{V}(K_{1}, \cdots, K_{n})=\frac{1}{n}\int_{S^{n-1}}\rho_{K_{1}}(u)\cdots\rho_{K_{n}}(u)du, $

where $du$ is the volume element of $S^{n-1}$.

Dual mixed volumes are the counterparts of mixed volumes. While the mixed volumes date back to Minkowski in the last century, the dual mixed volumes were only recently discovered. They play the roles in the study of cross-sections of convex bodies as the mixed volumes do for the study of projections of convex bodies. However, this duality is not at all trivial. One should read the book [2] for an excellent illustration. Dual mixed volumes are far from well understood. Their applications to the characterizations of intersection bodies and the solutions of the Busemann-Petty problem are very recent developments. See [7, 20, 21, 27].

Denote $\widetilde{V}(K, \cdots, K, L, \cdots, L)$ by $\widetilde{V_{i}}(K, L), $ where $K$ appears $n-i$ times and $L$ appears $i$ times. The dual quermassintegral $\widetilde{W}_{n-i}(K)$ is given by $\widetilde V_{i}(K, B_{n}).$ The importance of the dual quermassintegrals lies in the fact that the $(n-i)$th dual quermassintegral of a star body $K$ is proportional to the mean of the $i$-dimensional volumes of the slices of $K$ by the $i$-dimensional subspaces of $\mathbb{R}^{n}, $ that is

$ \widetilde{W}_{n-i}(K)=\frac{w_{n}}{w_{i}}\int_{G{(n, ~i)}}{\rm vol}_{i}(K\cap\xi)d\mu_{i}(\xi), $

where $G(n, ~i)$ is the Grassmann manifold of $i$-dimensional subspaces of $\mathbb{R}^{n}, $ and $\mu_{i}$ the Haar measure on $G(n, i), $ normalized by $\mu_{i}(G(n, i))=1.$ For $K\in\mathscr{S}, $ the intersection $K\cap u^{\perp}$ is a star body in $(n-1)$-dimensional space. Let $\widetilde{W}_{n-1-i}(K\cap u^{\perp})$ be the $(n-1-i)$ th dual quermassintegral of $K\cap u^{\perp}$ in $\mathbb{R}^{n-1}, $ which is called the dual $(n-1-i)$-girth of $K$ in the direction $u.$

A slight extension of the notation $\widetilde{V_{i}}(K, L)$ is

$ \widetilde{V_{r}}(K, L)=\frac{1}{n}\int_{S^{n-1}}\rho_{K}(u)^{n-r}\rho_{L}(u)^{r}du, r\in \mathbb{R}. $

The dual Minkowski inequalities state that

$ \widetilde{V}_{r}(K, L)^{n}\leq V(K)^{n-r}V(L)^{r}, {\rm if}\;\;r>0, \\\widetilde{V}_{r}(K, L)^{n}\geq V(K)^{n-r}V(L)^{r}, {\rm if}\;\; r<0, $

with equalities if and only if $K$ and $L$ are dilations of each other.

The intersection body $IK$ of a star body $K\in\mathscr{S}$ is defined as the centered body whose radial function is given by

$ \rho_{IK}(u)={\rm vol}_{n-1}(K\cap u^{\perp}), \quad u\in S^{n-1}, $

where $u^{\perp}$ is the $(n-1)$-dimensional subspace of $\mathbb{R}^{n}$ orthogonal to $u$. We will follow Zhang [25] to consider some generalizations of this definition.

The intersection body of order $i$ of a star body $K, $ $I_{i}K, $ is defined by

$ \rho_{{I_i}K}(u)=\widetilde{W}_{n-1-i}(K\cap u^{\perp}), u\in S^{n-1}. $

Hence, $IK=I_{n-1}K.$

The intersection body of order $i$ is closely related to the spherical Radon transform. For $f\in C(S^{n-1})$, the spherical Radon transform of $f$, $Rf$, is defined by

$ (Rf)(u)=\frac{1}{n-1}\int_{S^{n-1}\cap u^{\perp}} f(\nu)ds(\nu), $

where $ds$ is the volume element of $S^{n-1}\cap u^{\perp}.$ By applying the spherical Radon transform to the $i$th power of the radial function of a star body, we have

$ \begin{aligned}(R\rho_{K}^{i})(u)&=\frac{1}{n-1}\int_{S^{n-1}\cap u^{\perp}}\rho_{K}^{i}(\nu)ds(\nu)\\ &=\frac{w_{n-1}}{iw_{i}}\int_{G(n-1, ~i)}\int_{S^{n-2}\cap\xi}\rho_{K}^{i}(\nu)dS_{i-1}(B;~\nu)d\mu_{i}(\xi) \\&=\frac{w_{n-1}}{w_{i}}\int_{G(n-1, ~i)}{\rm vol}_{i}(K\cap u^{\perp})d\mu_{i}(\xi).\end{aligned} $

Hence, we have

$ \rho_{{I_i}K}(u)=(R\rho_{K}^{i})(u)=\widetilde{W}_{n-1-i}(K\cap u^{\perp}). $

When restricted to $C_{e}^{\infty}(S^{n-1}), $ the spherical Radon transform $R: C_{e}^{\infty}(S^{n-1})\rightarrow C_{e}^{\infty}(S^{n-1})$ is a continuous bijection (see Helgason [6, p.161]). It is also selfadjoint, i.e., for $f, g \in C(S^{n-1}), $ $(f, Rg)=(Rf, g).$

For $K\in\mathscr{S}_e$, we call the distribution $R^{-1}\rho_{K}$ the dual generating distribution of $K$, denoted by $\widetilde{\mu}_{K}.$

Let $\phi$ be a function from the Schwartz space $\mathcal{S}$ of rapidly decreasing infinitely differentiable functions on $\mathbb{R}^{n}.$ We define the Fourier transform of $\phi$ by

$ \widehat{\phi}(\xi)=\int_{\mathbb{R}^{n}}\phi(x)e^{-i(x, ~\xi)}dx, \xi\in\mathbb{R}^{n}. $

The Fourier transform of a distribution $f$ is defined by $\langle\widehat{f}, \phi\rangle=\langle f, \widehat{\phi}\rangle$ for every test function $\phi\in \mathcal{S}.$ We say that a distribution is positively definite if its Fourier transform is a positive distribution, in the sense that $\langle\widehat{f}, ~\phi\rangle\geq 0$ for every non-negative test function $\phi.$

For the applications of Fourier transform to convex geometry, one can refer the papers, for example, [4, 5, 7, 8, 10, 25] and the book [9].

A well-known theorem of Lewis [11] will be used in the proof of Theorem 1. We presented it here in the following form proved in [17] (see Theorem 8.2) by Lutwak, Yang and Zhang.

Lemma 1   If $\ell$ is an $n$-dimensional subspace of $L_{p}$, then $\ell$ is isometric to the Banach space $(\mathbb{R}^{n}, \|\centerdot\|)$ where the norm $\|\centerdot\|$ can be represented by a finite Borel measure, $\mu$, such that for all $x\in \mathbb{R}^{n}$,

$ \|x\|=(\int_{S^{n-1}}\mid x\centerdot v\mid^{_{p}}d\mu(v))^{\frac{1}{p}}, $

and

$ \mid x \mid=(\int_{S^{n-1}}\mid x\centerdot v\mid^{_{2}}d\mu(v))^{\frac{1}{2}}. $

Theorem 1   Let $K\in \mathcal{L}_{p}$, $1\leq p < \infty, $ and $L$ be a star body. Then

$ \widetilde{V}_{-p}(L, ~K)=\frac{(n+p)V(L)}{n}\int_{S^{n-1}}h^{p}_{\Gamma_{p}L}(v)d\mu_{K}(v), $

where $\mu_{K}$ is a finite Borel measure on $S^{n-1}.$ Specially, if $L$ is the unit ball $B_{n}, $ then

$ \widetilde{V}_{-p}(B_{n}, ~K)=n\omega_{n}c_{n-2, ~p}. $

Proof   According to Lemma 1, we know that if $(\mathbb{R}^{n}, \|\centerdot\|_{K})$ is a subspace of $L_{p}$, $p \geq 1, $ then there exists a position of the body $K$ and a finite Borel measure $\mu_{K}$ on $S^{n-1}$ such that for all $x \in \mathbb{R}^{n}, $

$ \|x\|^{p}_{K}=\int_{S^{n-1}}\mid x\centerdot v\mid^{_{p}}d\mu_{K}(v), $

and

$ \mid x \mid^{2}=\int_{S^{n-1}}\mid x\centerdot v\mid^{_{2}}d\mu_{K}(v). $

(*)

From the definition of dual mixed volume and Fubini's theorem, we have

$ \begin{aligned} \widetilde{V}_{-p}(L, ~K) &=\frac{1}{n}\int_{S^{n-1}}\rho^{n+p}_{L}(u)\rho^{-p}_{K}(u)du\\ &=\frac{1}{n}\int_{S^{n-1}}\rho^{n+p}_{L}(u)(\int_{S^{n-1}}\mid u\centerdot v\mid^{_{p}}d\mu_{K}(v))du\\ &=\frac{1}{n}\int_{S^{n-1}}(\int_{S^{n-1}}\rho^{n+p}_{L}(u)\mid u\centerdot v\mid^{_{p}}du)d\mu_{K}(v)\\ &=\frac{(n+p)V(L)}{n}\int_{S^{n-1}}(\frac{1}{(n+p)V(L)}\int_{S^{n-1}}\rho^{n+p}_{L}(u)\mid u\centerdot v\mid^{_{p}}du)d\mu_{K}(v)\\ &=\frac{(n+p)V(L)}{n}\int_{S^{n-1}}h^{p}_{\Gamma_{p}L}(v)d\mu_{K}(v). \end{aligned} $

When $L$ is the unit ball $B_{n}, $

$ h^{p}_{\Gamma_{p}B_{n}}(v) =\frac{1}{(n+p)\omega_{n}}\int_{S^{n-1}}\rho^{n+p}_{B_{n}}(u)\mid u\centerdot v\mid^{_{p}}du =\frac{1}{(n+p)\omega_{n}}\int_{S^{n-1}}\mid u\centerdot v\mid^{_{p}}du =\frac{n\omega_{_{n}}c_{_{n-2, p}}}{(n+p)\omega_{n}}. $

Finally, we only need to compute the measure of $S^{n-1}$ with respect to $\mu_{K}.$ Let $\sigma$ be the normalized Haar measure on the sphere. Integrating equation (*) with respect to $\sigma$ we get

$ \begin{eqnarray*}1&=&\int_{S^{n-1}}\mid x \mid^{2}d\sigma(x)=\int_{S^{n-1}}\int_{S^{n-1}}\mid x\centerdot v\mid^{_{2}}d\mu_{K}(v)d\sigma(x)\\ &=&\int_{S^{n-1}}\mid x_{1} \mid^{2}d\sigma(x)\cdot \int_{S^{n-1}}d\mu_{K}(v)=\frac{1}{n}\int_{S^{n-1}}d\mu_{K}(v).\end{eqnarray*} $

Hence, when $L=B_{n}, $ we have $\widetilde{V}_{-p}(B_{n}, ~K)=n\omega_{n}c_{n-2, ~p}.$

This completes the proof.

From Theorem 1, we can get the following property immediately.

Corollary 1   Let $K\in \mathcal{L}_{p}$, $1\leq p < \infty.$ Then

$ \int_{S^{n-1}}h^{p}_{\Gamma_{p}K}(v)d\mu_{K}(v)=\frac{n}{n+p}, $

where $\mu_{K}$ is a finite Borel measure on $S^{n-1}.$

Theorem 2   Let $K\in \mathcal{L}_{p}$, $1\leq p < \infty.$ Then for any star bodies $L$ and $M, $ we have

$ \Gamma_{p}L \subseteq \Gamma_{p}M \Longrightarrow \frac{\widetilde{V}_{-p}(L, ~K)}{V(L)}\leq \frac{\widetilde{V}_{-p}(M, ~K)}{V(M)}. $

Proof   From the proof of Theorem 1, we have

$ \begin{eqnarray*}&\frac{\widetilde{V}_{-p}(L, ~K)}{V(L)}=\frac{n+p}{n}\int_{S^{n-1}}h^{p}_{\Gamma_{p}L}(v)d\mu_{K}(v), \\ &\frac{\widetilde{V}_{-p}(M, ~K)}{V(M)}=\frac{n+p}{n}\int_{S^{n-1}}h^{p}_{\Gamma_{p}M}(v)d\mu_{K}(v).\end{eqnarray*} $

If $\Gamma_{p}L \subseteq \Gamma_{p}M, $ then we have $h_{\Gamma_{p}L}\leq h_{\Gamma_{p}M}, $ Therefore, the following inequality

$ \frac{\widetilde{V}_{-p}(L, ~K)}{V(L)}\leq \frac{\widetilde{V}_{-p}(M, ~K)}{V(M)} $

holds.

This completes the proof.

Corollary 2   Let $K\in \mathcal{L}_{p}$, $1\leq p < \infty.$ Then for any star body $L, $ we have

$ \Gamma_{p}L \subseteq \Gamma_{p}K \Longrightarrow V(L)\leq V(K). $

Proof   Let $M=K$ in Theorem 2. Then $\frac{\widetilde{V}_{-p}(L, ~K)}{V(L)}\leq 1.$ By the dual Minkowski inequality, we have

$ \frac{\widetilde{V}_{-p}(L, ~K)}{V(L)}\geq \frac{V(L)^{\frac{n+p}{n}}V(K)^{\frac{-p}{n}}}{V(L)}. $

Therefore, $V(L)\leq V(K).$

This completes the proof.

Remark   In [5] (see Theorem 4.11 and Corollary 4.13), the authors proved the similar results as our Theorem 2 and Corollary 2, respectively, when $L$ and $M$ are centered star bodies.

Theorem 3   Let $K\in\mathscr{S}_e$. If $K$ is an intersection body, then for all $L_{1}$, $L_{2}$, $M_{1}$, $M_{2}\in \mathscr{S}_e$,

$ \frac{\widetilde{V}(L_{1}\widetilde{+}L_{2}, ~i;~K, ~1;~B_{n})}{\widetilde{V}(M_{1}\widetilde{+}M_{2}, ~i;~K, ~1;~B_{n})}\leq \sum\limits_{k=0}^{i}\max\limits_{u\in{S}^{n-1}} \frac{\widetilde{V}(L_{1}\cap u^{\perp}, ~k;~L_{2}\cap u^{\perp}, ~i-k;~B_{n}, ~n-i)}{\widetilde{V}(L_{1}\cap u^{\perp}, ~k;~L_{2}\cap u^{\perp}, ~i-k;~B_{n}, ~n-i)}. $

Proof   Suppose $K$ is an intersection body with dual generating measure $\widetilde{\mu_{k}}.$Then

$ \begin{array}{l}\frac{\widetilde{V}(L_{1}\widetilde{+}L_{2}, ~i;~K, ~1;~B_{n})}{\widetilde{V}(M_{1}\widetilde{+}M_{2}, ~i;~K, ~1;~B_{n})}&=\frac{({\rho_{L_{1}\widetilde{+}L_{2}}^{i}}, ~\rho_k)}{({\rho_{M_{1}\widetilde{+}M_{2}}^{i}}, ~\rho_k)} =\frac{((\rho_{L_1}+\rho_{L_2})^{i}, ~\rho_{K})}{((\rho_{M_1}+\rho_{M_2})^{i}, ~\rho_{K})} \\&=\frac{(R(\rho_{L_1}+\rho_{L_2})^{i}, ~R^{-1}\rho_{K})}{(R(\rho_{M_1}+\rho_{M_2})^{i}, ~R^{-1}\rho_{K})}\\ &=\frac{(R\sum\limits_{k=0}^{i}{i \choose k}\rho_{L_1}^{k}\rho_{L_2}^{i-k}, ~R^{-1}\rho_{K})} {(R\sum\limits_{k=0}^{i}{i \choose k}\rho_{M_1}^{k}\rho_{M_2}^{i-k}, ~R^{-1}\rho_{K})}\\ &=\frac{\sum\limits_{k=0}^{i}{i \choose k}\int_{S^{n-1}}R(\rho_{L_1}^{k}\rho_{L_2}^{i-k})d\widetilde{u_k}(u)} {\sum\limits_{k=0}^{i}{i \choose k}\int_{S^{n-1}}R(\rho_{M_1}^{k}\rho_{M_2}^{i-k})d\widetilde{u_k}(u)} \\&\leq\sum\limits_{k=0}^{i}\max\limits_{u\in{S}^{n-1}}\frac{R(\rho_{L_1}^{k}\rho_{L_2}^{i-k})}{R(\rho_{M_1}^{k}\rho_{M_2}^{i-k})} \\&=\sum\limits_{k=0}^{i}\max\limits_{u\in{S}^{n-1}}\frac{\int_{{S^{n-1}}\cap u^{\perp}}\rho_{L_1}^{k}\rho_{L_2}^{i-k}ds(\nu)} {\int_{{S^{n-1}}\cap u^{\perp}}\rho_{M_1}^{k}\rho_{M_2}^{i-k}ds(\nu)} \\&=\sum\limits_{k=0}^{i}\max\limits_{u\in{S}^{n-1}}\frac{\int_{S^{n-1}}\rho_{L_1\cap u^{\perp}}^{k}\rho_{L_2\cap u^{\perp}}^{i-k}ds(\nu)} {\int_{S^{n-1}}\rho_{M_1\cap u^{\perp}}^{k}\rho_{M_2\cap u^{\perp}}^{i-k}ds(\nu)} \\&=\sum\limits_{k=0}^{i}\max\limits_{u\in{S}^{n-1}}\frac{\widetilde{V}(L_{1}\cap u^{\perp}, ~k;~L_{2}\cap u^{\perp}, ~i-k;~B_{n}, ~n-i)}{\widetilde{V}(M_{1}\cap u^{\perp}, ~k;~M_{2}\cap u^{\perp}, ~i-k;~B_{n}, ~n-i)}.\end{array}. $

This completes the proof.

Corollary 3   Let $K\in \mathscr{S}_e$. If $K$ is an intersection body. Then for all $L, $ $M\in \mathscr{S}_e, $

$ \frac{\widetilde{V}(L, ~i;~K, ~1;~B)}{\widetilde{V}(M, ~i;~K, ~1;~B)}\leq \max\limits_{u\in{S}^{n-1}} \frac{\widetilde{W}_{n-1-i}(L\cap u^{\perp})}{\widetilde{W}_{n-1-i}(M\cap u^{\perp})}. $

Proof   From Theorem 3, we have

$ \begin{aligned}&\frac{\widetilde{V}(L, ~i;~K, ~1;~B)}{\widetilde{V}(M, ~i;~K, ~1;~B)}=\frac{(\rho_{L}^{i}, ~\rho_{K})}{(\rho_{M}^{i}, ~\rho_{K})} =\frac{(R\rho_{L}^{i}, ~R^{-1}\rho_{K})}{(R\rho_{M}^{i}, ~R^{-1}\rho_{K})}\\ =&\frac{\int_{u\in{S^{n-1}}}\rho_{{I_i}L}(u)d\widetilde{u_k}(u)}{\int_{u\in{S^{n-1}}}\rho_{{I_i}M}(u)d\widetilde{u_k}(u)} \leq\max\limits_{u\in{S}^{n-1}}\frac{\rho_{{I_i}L}(u)}{\rho_{{I_i}L}(u)} =\max\limits_{u\in{S}^{n-1}}\frac{\widetilde{W}_{n-1-i}(L\cap u^{\perp})}{\widetilde{W}_{n-1-i}(M\cap u^{\perp})}.\end{aligned} $

This completes the proof.

To prove the Theorem 4, we will need the following version of Parseval's formula on the sphere proved in [7].

Lemma 2    If $K$ and $L$ are origin symmetric infinitely smooth convex bodies in $\mathbb{R}^{n}$ and $0< p<n, $ then $(\|x\|^{-p}_{K})^{\wedge}$ and $(\|x\|^{-n+p}_{L})^{\wedge}$ are continuous functions on $S^{n-1}$ and

$ \int_{S^{n-1}}(\|x\|^{-p}_{K})^{\wedge}(\xi)(\|x\|^{-n+p}_{L})^{\wedge}(\xi)d\xi=(2\pi)^{n}\int_{S^{n-1}}\|x\|^{-p}_{K} \|x\|^{-n+p}_{L}dx. $

The concept of embedding of a normed spaces in $L_{-p}$ with $0<p<n$ was introduced in [8] by Koldobsky. It was also proved that, as for positive $p, $ there is a Fourier analytic characterization for such embeddings, namely a space $(\mathbb{R}^{n}, \|\centerdot\|)$ embeds in $L_{-p}$ if and only if the Fourier transform of $\|\centerdot\|^{-p}$ is a positive distribution in $\mathbb{R}^{n}.$ In [10], the unit balls of such spaces are called $L_{-p}$-balls or $p$-intersection bodies.

Theorem 4  Let $K$ and $L$ be origin symmetric infinitely smooth convex bodies in $\mathbb{R}^{n}.$ If $L$ is a $L_{-p}$-ball, $0< p < n, $ then $\widetilde{V}_{p}(K, ~L)\leq C V(K)^{\frac{p}{n}}\max\limits_{\xi \in S^{n-1}}(\|x\|^{-n+p}_{L})^{\wedge}(\xi), $ where

$ C=\frac{\Gamma(\frac{n-p}{2})}{2^{p}\pi^{\frac{n}{2}}n^{\frac{n-p}{n}}\Gamma(\frac{p}{2})}\alpha^{\frac{n-p}{n}}_{n-1}. $

Proof   According to the definition of dual mixed volume and Lemma 2, we have

$ \begin{aligned} \widetilde{V}_{p}(K, ~L) &=\frac{1}{n}\int_{S^{n-1}} \rho^{p}_{K}(u)\rho^{n-p}_{L}(u)du\\ &=\frac{1}{n}\int_{S^{n-1}} \|x\|^{-p}_{K} \|x\|^{-n+p}_{L}dx\\ &=\frac{1}{(2\pi)^{n}n}\int_{S^{n-1}}(\|x\|^{-p}_{K})^{\wedge}(\xi)(\|x\|^{-n+p}_{L})^{\wedge}(\xi)d\xi. \end{aligned} $

If $L$ is a $L_{-p}$-ball, then $(\|x\|^{-p}_{K})^{\wedge}(\xi)\geq 0, $ therefore

$ \widetilde{V}_{p}(K, ~L)\leq \frac{1}{(2\pi)^{n}n}\int_{S^{n-1}}(\|x\|^{-p}_{K})^{\wedge}(\xi)d\xi \cdot \max\limits_{\xi \in S^{n-1}}(\|x\|^{-n+p}_{L})^{\wedge}(\xi) $

Using that (see [3], p.192):

$ (\mid x \mid^{-n+p}_{2})^{\wedge}(\xi)=2^{p}\pi^{\frac{n}{2}}\frac{\Gamma(\frac{p}{2})}{\Gamma(\frac{n-p}{2})}\mid \xi \mid^{-p}_{2}, $

and applying Parseval's formula again, then Hölder's inequality, we have

$ \begin{array}{l} \widetilde{V}_{p}(K, ~L) &\leq \frac{2^{-p}\pi^{\frac{-n}{2}}}{(2\pi)^{n}n}\frac{\Gamma(\frac{n-p}{2})}{\Gamma(\frac{p}{2})}\int_{S^{n-1}}(\|x\|^{-p}_{K})^{\wedge}(\xi)(\mid x \mid^{-n+p}_{2})^{\wedge}(\xi)d\xi \times \max\limits_{\xi \in S^{n-1}}(\|x\|^{-n+p}_{L})^{\wedge}(\xi)\\ &=\frac{2^{-p}\pi^{\frac{-n}{2}}}{n}\frac{\Gamma(\frac{n-p}{2})}{\Gamma(\frac{p}{2})}\int_{S^{n-1}}\|x\|^{-p}_{K}dx \cdot \max\limits_{\xi \in S^{n-1}}(\|x\|^{-n+p}_{L})^{\wedge}(\xi)\\ &\leq \frac{2^{-p}\pi^{\frac{-n}{2}}}{n}\frac{\Gamma(\frac{n-p}{2})}{\Gamma(\frac{p}{2})}(\int_{S^{n-1}}\|x\|^{-n}_{K}dx)^{\frac{p}{n}}\cdot (\alpha_{n-1})^{\frac{n-p}{n}}\cdot \max\limits_{\xi \in S^{n-1}}(\|x\|^{-n+p}_{L})^{\wedge}(\xi)\\ &=CV(K)^{\frac{p}{n}}\cdot \max\limits_{\xi \in S^{n-1}}(\|x\|^{-n+p}_{L})^{\wedge}(\xi), \end{array} $

where

$ C=\frac{\Gamma(\frac{n-p}{2})}{2^{p}\pi^{\frac{n}{2}}n^{\frac{n-p}{n}}\Gamma(\frac{p}{2})}\alpha^{\frac{n-p}{n}}_{n-1}. $

This completes the proof.

When $K=L, $ we can get the following result from Theorem 4 immediately.

Corollary 5   Let $K$ be an origin symmetric infinitely smooth convex bodies in $\mathbb{R}^{n}.$ If $K$ is a $L_{-p}$-ball, $0< p < n, $ then

$ V(K)^{\frac{n-p}{n}}\leq C \max\limits_{\xi \in S^{n-1}}(\|x\|^{-n+p}_{K})^{\wedge}(\xi), $

where

$ C=\frac{\Gamma(\frac{n-p}{2})}{2^{p}\pi^{\frac{n}{2}}n^{\frac{n-p}{n}}\Gamma(\frac{p}{2})}\alpha^{\frac{n-p}{n}}_{n-1}. $

From Theorem 3.8 in [9], we have

$ (\|x\|^{-n+1}_{L})^{\wedge}(\xi)=\pi(n-1){{\rm Vol}}_{n-1}(K\cap\xi^{\perp}). $

So according to Corollary 5, when $p=1, $ we can get

Corollary 6   Let $K$ be an origin symmetric infinitely smooth convex bodies in $\mathbb{R}^{n}.$ If $K$ is a $L_{-1}$-ball, then

$ V(K)^{\frac{n-1}{n}}\leq \frac{\Gamma (\frac{n-1}{2})}{2^{\frac{1}{n}}\pi(n\Gamma(\frac{n}{2}))^{\frac{n-1}{n}}}\max\limits_{\xi \in S^{n-1}}{\rm vol}_{n-1}(K \cap \xi^{\perp} ). $

Corollary 6 is related to the maximal slice conjecture. This conjecture states that for any origin symmetric convex body there exists an universal constant $c>0$ such that

$ V(K)^{\frac{n-1}{n}}\leq c\max\limits_{\xi \in S^{n-1}}{\rm vol}_{n-1}(K \cap \xi^{\perp} ). $

For related problems, see [1, 22].