A convex body $K$ (i.e., a compact, convex subset with nonempty interior) in Euclidean $n$-space $\mathbb{R}^{n}$, is determined by its support function, $h(K,\cdot):S^{n-1}\mapsto\mathbb{R}^{n}$, on the unit sphere $S^{n-1}$, where $h(K,u)=\max\{u\cdot x|x\in K\}$ and where $u\cdot x$ denotes the standard inner product of $u$ and $x$. The projection body, $\Pi K$, of $K$ is the convex body whose support function, for $u\in S^{n-1}$, is given by $ h(\Pi K,u)= {\rm vol}_{n-1}(K|u^{\bot}), $ where vol$_{n-1}$ denotes the $(n-1)$-dimensional volume and $K|u^{\bot}$ denotes the image of the orthogonal projection of $K$ onto the codimension $1$ subspace orthogonal to $u$.

Projection bodies were introduced by Minkowski at the beginning of the previous century in connection with Cauchy's surface area formula. Since 1980s, projection bodies received considerable attention. An important unsolved problem regarding projection bodies is Schneider's projection problem (see [7]): what is the least upper bound, as $K$ ranges over the class of origin-symmetric convex bodies in $\mathbb{R}^{n}$, of the affine-invariant ratio

$\begin{aligned} \left [V(\Pi K)/V(K)^{n-1} ]^{\frac{1}{n}}, \right. \end{aligned}$

where $V$ is used to denote the $n$-dimensional volume.

An effective tool to study Schneider's projection problem is the cone volume functional $U$ introduced by Lutwak, Yang and Zhang [1]: if $P$ is a convex polytope in $\mathbb{R}^{n}$ which contains the origin $o$ in its interior, then define $U(P)$ through the formula $ U(P)^{n}=\frac{1}{n^{n}}\sum\limits_{u_{i_{1}}\bigwedge\cdots\bigwedge u_{i_{n}}\neq0}h_{i_{1}}\cdots h_{i_{n}}a_{i_{1}}\cdots a_{i_{n}}, $ where $u_{i_{1}},\cdots, u_{i_{N}}$ are the outer normal unit vectors to the corresponding facets $F_i$ of $F_{1},\cdots, F_{N}$ of $P$, and the facet with outer normal vector $u_{i}$ has area (i.e., $(n-1)$-dimensional volume) $a_{i}$ and distance $h_{i}$ from the origin.

Let $V_{i}=\frac{1}{n}h_{i}a_{i}$. Then $V_{i}$ is the volume of the cone conv$(o, F_{i})$, and

$ U(P)^{n}=\sum\limits_{u_{i_{1}}\bigwedge\cdots\bigwedge u_{i_{n}}\neq0}V_{i_{1}}\cdot\cdot\cdot V_{i_{n}},$

obviously the functional $U$ is centro-affine invariant, i.e.,

$ U({\varnothing }P)=U(P), \forall{\varnothing }\in SL(n), $

since $V(P)=\frac{1}{n}\sum\limits^{N}_{i=1}a_{i}h_{i}$, it follows that $U(P)/V(P)\leq1$.

By the way, we observe the cone volume functional $U$ has strong connection with the cone measure: for every star-shaped body $K\subseteq\mathbb{R}^{n}$, the cone measure of a subset $A$ and vertex $o$. The cone measure appears in the Gromov-Milman theorem [9] on the concentration of Lipschitz functions on uniformly convex bodies. In [8], Anor established the precise relation between the surface measure and cone measure on the sphere of $l^{n}_{p}$.

One fundamental, but still remaining open extremum problem, on the ratio of $U$ to $V$ is posed by Lutwak, Deane, and Zhang [1].

Conjecture  If $P$ is a convex polytope in $\mathbb{R}^{n}$ with its centroid at the origin, then

$ \frac{U(P)}{V(P)}\geq\frac{(n!)^{1/n}}{n} $

with equality when and only when $P$ is a parallelotope.

The first progress on LYZ's conjecture was due to He, Leng and Li [2]. They proved that the conjecture is true when restricted to the class of origin-symmetric convex polytopes.

Theorem 1.1  Suppose that $P\subseteq\mathbb{R}^{n}$ is an origin-symmetric convex polytope, then

$ \frac{U(P)}{V(P)}\geq\frac{(n!)^{1/n}}{n}$

with equality when and only when $P$ is a parallelotope.

Lutwak, Yang and Zhang presented a version of Schneider's conjecture that has an affirmative answer. They proved the following important theorems.

Theorem 1.2 Suppose $K$ is an origin-symmetric convex polytope in $\mathbb{R}^{n}$, then

$ \frac{V(\Pi K)}{U(K)^{\frac{n}{2}}V(K)^{\frac{n}{2}-1}}\leq2^{n}(\frac{n^{n}}{n!})^{\frac{1}{2}} $

with equality when and only when $K$ is a parallelotope.

By Theorems 1.1 and 1.2, the following theorem is obtained.

Theorem 1.3  Suppose $K$ is an origin-symmetric convex polytope in $\mathbb{R}^{n}$, then

$ \frac{V(\Pi K)}{U(K)^{n-1}}\leq2^{n}\frac{n^{n-1}}{(n!)^{(n-1)/n}} $

with equality when and only when $K$ is a parallelotope.

Theorem 1.3 can be seen as a modified version of Schneider's projection conjecture.

This paper is devoted to the study of LYZ's conjecture. We give another answer to the cone volume inequality in origin-symmetric convex bodies.

Theorem 1.4  Suppose that $P\subset\mathbb{R}^{j}\times\mathbb{R}^{n-j}$ is an origin-symmetric convex body with interior points. $V(P)$ is the volume of $P$, $K$ is a convex cone, the vertex of which is at the origin, then $ V(K)\leq\frac{j}{n}V(P). $

Definition 2.1 If $P\subset\mathbb{R}^{j}\times\mathbb{R}^{n-j},1\leq j\leq n-1$, is an origin-symmetric convex body with interior points. Suppose $(x^{o},y^{o})$ is the centroid of $P$, $x^{o}\in\mathbb{R}^{j},y^{o}\in\mathbb{R}^{n-j}$, $D=K|\mathbb{R}^{j}$, $L\subset\mathbb{R}^{n}$ is a $j$-dimensional subspace, and $f(x)={\rm vol}_{n-j}(P\bigcap(L^{\bot}+x)), x\in D$, $u_{x}$ is the outer normal unit vector of $x$ on $\partial D$. For convex cone $K$, the vertex of which is at the origin, then define the volume of $K$ by

$ V(K)=\frac{1}{n}\int_{\partial D}\langle(x-x^{o}),u_{x}\rangle f(x)dS(x)=\frac{1}{n}\int_{\partial D}(x-x^{o})f(x)d\textbf{S}(x). $

Theorem 2.1  Let $P\in\mathbb{R}^{j}\times\mathbb{R}^{n-j},1\leq j\leq n-1$, be an origin-symmetric smooth convex body with interior points, $V(P)$ is the volume of $P$, then

$\begin{equation} \label{eq:1} V(K)\leq\frac{j}{n}V(P). \end{equation}$

(2.1)

Proof Suppose $D=K|\mathbb{R}^{j}$, $L\subset\mathbb{R}^{n}$ is a $j$-dimensional subspace, and

$ f(x)={\rm vol}_{n-j}(P\bigcap(L^{\bot}+x)),x\in D. $

Let $(x^{o},y^{o})$ is the centroid of $P, x^{o}\in\mathbb{R}^{j},y^{o}\in\mathbb{R}^{n-j},u_{x}$ is the outer normal unit vector of $x$ on $\partial D$, then

$ V(K)=\frac{1}{n}\int_{\partial D}\langle(x-x^{o}),u_{x}\rangle f(x)dS(x)=\frac{1}{n}\int_{\partial D}(x-x^{o})f(x)d\textbf{S}(x). $

According to the Gauss formula

$\begin{aligned} V(K) &=\frac{1}{n}\int_{\partial D}\begin{pmatrix}(x_{1}-x^{o}_{1})f(x_{1},\cdots,x_{j}) \\ \vdots\\(x_{j}-x^{o}_{j})f(x_{1},\cdots, x_{j}) \end{pmatrix}d\textbf{S}(x) \\&=\frac{1}{n}\int_{D}[f'_{1}\cdot(x_{1}-x^{o}_{1})+\cdots+f'_{j}\cdot(x_{j}-x^{o}_{j})+j\cdot f]d\sigma \\&=\frac{j}{n}\int_{D}fd\sigma+\frac{1}{n}\int_{D}({\rm grad}f,(x-x^{o}))d\sigma \\&=\frac{j}{n}V(P)+\frac{1}{n}\int_{D}({\rm grad}f,(x-x^{o}))d\sigma. \end{aligned}$

The direction of grad$g$ varies with the direction of contour surface, and is directed from the low value to the high value, therefore the contour surface and $(x-x^{o})$ form an obtuse angle, obviously $ \displaystyle\int_{D}({\rm grad}f,(x-x^{o}))d\sigma\leq0, $ so that $ V(K)\leq\frac{j}{n}V(P). $

Lemma 2.2  Let $P\subset\mathbb{R}^{n}$ be a convex body and $L\subset\mathbb{R}^{n}$ be a $j$-dimensional subspace, $1\leq j\leq n-1$. If $ f:L\mapsto\mathbb{R},f(x)={\rm vol}_{n-j}(P\bigcap(L^{\bot}+x)), $ then $f^{\frac{1}{n-j}}$ is concave on $P\mid L$.

From [6], we know that continuous convex function on a Banach space can be approximated by a smooth convex function, it follows immediately that

Lemma 2.3  Let $f^{\frac{1}{n-j}}$ be a concave function on $D$, for any $\varepsilon>0$, there exists a $M>0$ and a smooth concave function $g^{\frac{1}{n-j}}$, $\mid f^{\frac{1}{n-j}}-g^{\frac{1}{n-j}}\mid<\frac{\varepsilon}{M}$, then $f$ approximates $g$.

Proof  Suppose $\widetilde{f}=f^{\frac{1}{n-j}},\widetilde{g}=g^{\frac{1}{n-j}}$, from the Lagrange's mean value theorem, it follows that

$ \mid\widetilde{f}^{n-j}-\widetilde{g}^{n-j}\mid=\mid(n-j)u^{n-j-1}(\widetilde{f}-\widetilde{g})\mid<\frac{\varepsilon}{M}, $

let $f\leq u\leq g$ (or $g\leq u\leq f$), the function $u$ is bounded, then $ \mid f-g\mid=\mid\widetilde{f}^{n-j}-\widetilde{g}^{n-j}\mid<\varepsilon, $ so that $f$ approximates $g$.

Proof of Theorem 1.4  Suppose $D=K|\mathbb{R}^{j}$, $L\subset\mathbb{R}^{n}$ is a $j$-dimensional subspace, and $f(x)={\rm vol}_{n-j}(P\bigcap(L^{\bot}+x)),x\in D$.

Let $(x^{o},y^{o})$ is the centroid of $P$, $x^{o}\in\mathbb{R}^{j},y^{o}\in\mathbb{R}^{n-j}$, $u_{x}$ is the outer normal unit vector of $x$ on $\partial D$, and

$\begin{aligned} V(K)&=\frac{1}{n}\int_{\partial D}((x-x^{o}),u_{x})f(x)dS(x) \\&=\frac{1}{n}\int_{\partial D}(x-x^{o})|f-g+g|d\textbf{S}(x) \\&\leq\frac{1}{n}\int_{\partial D}(x-x^{o})|f-g|d\textbf{S}(x)+\frac{1}{n}\int_{\partial D}(x-x^{o})gd\textbf{S}(x) \\&=\frac{1}{n}\int_{\partial D}(x-x^{o})\varepsilon d\textbf{S}(x)+\frac{1}{n}\int_{\partial D}(x-x^{o})gd\textbf{S}(x) \\&=\frac{1}{n}\int_{\partial D}(x-x^{o})gd\textbf{S}(x)+\frac{j}{n}\int_{D}\varepsilon d\sigma. \end{aligned}$

According to the Gauss formula

$\begin{aligned} V(K) &=\frac{1}{n}\int_{\partial D}\begin{pmatrix}(x_{1}-x^{o}_{1})g(x_{1},\cdots,x_{j}) \\ \vdots\\(x_{j}-x^{o}_{j})g(x_{1},\cdots,x_{j}) \end{pmatrix}d\textbf{S}(x)+\frac{j}{n}\int_{D}\varepsilon d\sigma \\&=\frac{1}{n}\int_{D}[g'_{1}\cdot(x_{1}-x^{o}_{1})+\cdots+g'_{j}\cdot(x_{j}-x^{o}_{j})+j\cdot g]d\sigma+\frac{j}{n}\int_{D}\varepsilon d\sigma \\&=\frac{j}{n}\int_{D}g d\sigma+\frac{1}{n}\int_{D}({\rm grad}g,(x-x^{o}))d\sigma+\frac{j}{n}\int_{D}\varepsilon d\sigma \\&=\frac{j}{n}\int_{D}(g+\varepsilon)d\sigma+\frac{1}{n}\int_{D}({\rm grad}g,(x-x^{o}))d\sigma. \end{aligned}$

The direction of grad$g$ varies with the direction of contour surface, and is directed from the low value to the high value, therefore the contour lines and $(x-x^{o})$ form an obtuse angle, obviously

$ \int_{D}({\rm grad}g,(x-x^{o}))d\sigma\leq0, $

so that $ V(K)\leq\frac{j}{n}\int_{D}(g+\varepsilon)d\sigma, $ since $|f-g|<\varepsilon$, then $ V(K)\leq\frac{j}{n}V(P). $

This completes the proof.

For origin-symmetric convex polytopes, the first progress was due to He-Leng-Li [2]. They gave an affirmative answer to the LYZ's conjecture in $\mathbb{R}^{n}$. They proved the inequality

$\sum\limits_{u_{i_{1}}\bigwedge\cdot\cdot\cdot\bigwedge u_{i_{j}}\bigwedge u_{i_{k}}=0}V_{i_{k}}\leq\frac{j}{n}V(P),1\leq j\leq n-1. $

And from the definition of $U(P)$, the proof of Theorem 1.1 was obtained in [2]. To make the paper self-contained, we present it here.

Proof of Theorem 1.1 

$\begin{aligned} U(P)^{n}=&\sum\limits_{u_{i_{1}}\bigwedge\cdots\bigwedge u_{i_{n}}\neq0}V_{i_{1}}\cdots V_{i_{n}}\\ =&\sum\limits_{u_{i_{1}}\bigwedge\cdots\bigwedge u_{i_{n-1}}\neq0}V_{i_{1}}\cdots V_{i_{n-1}}(V-\sum\limits_{u_{i_{1}}\bigwedge\cdots\bigwedge u_{i_{n-1}}\bigwedge u_{i_{k}}=0}V_{i_{k}}) \\\geq&\sum\limits_{u_{i_{1}}\bigwedge\cdots\bigwedge u_{i_{n-1}}\neq0}V_{i_{1}}\cdots V_{i_{n-1}}(V-\frac{n-1}{n}V) \\&\vdots \\=&\frac{(n-2)!}{n^{n-2}}V^{n-2}\sum\limits_{u_{i_{1}}u_{i_{2}}\neq0}V_{i_{1}}V_{i_{2}} \\=&\frac{(n-2)!}{n^{n-2}}V^{n-2}\sum\limits_{u_{i_{1}}\neq0}V_{i_{1}}(V-\sum\limits_{u_{i_{1}}u_{i_{k}}=0}V_{k}) \\\geq&\frac{(n-1)!}{n^{n-1}}V^{n-1}\sum\limits_{u_{i_{1}}\neq0}V_{i_{1}} =\frac{n!}{n^{n}}V^{n}, \end{aligned}$

that is,

$ U(P)\geq\frac{(n!)^{1/n}}{n}V(P),$

where the equality holds when and only when $P$ is a parallelotope.