标题： 一种离散处理张的投影不等式的方法 显示英文标题

标题： A discrete approach to Zhang's projection inequality

摘要： 在本文中，我们将提供一个新的证明，说明对于任何凸体$K\subseteq\R^n$ $$ \frac{{{2n}\choose{n}}}{n^n}n\int_0^\infty r^{n-1}\vol_n(K\cap(re_n+K))dr\leq\frac{(\vol_n(K))^{n+1}}{(\vol_{n-1}(P_{e_n^\perp}(K)))^n}, $$ ，其中$(e_i)_{i=1}^n$表示$\R^n$中的规范正交基，$P_{e_n^\perp}(K)$表示$K$在与$e_n$正交的线性超平面上的正交投影，而$\vol_k$表示$k$维勒贝格测度。 这个不等式由Gardner和Zhang证明，并且它蕴含了Zhang的不等式。我们将使用我们的新方法来处理这个不等式，以证明该不等式的离散类似形式以及其等价形式，其中我们将考虑格点计数器测度而不是勒贝格测度，并表明从这些离散类似形式中我们可以恢复上述不等式，因此也得到Zhang的不等式。 显示英文摘要

摘要： In this paper we will provide a new proof of the fact that for any convex body $K\subseteq\R^n$ $$ \frac{{{2n}\choose{n}}}{n^n}n\int_0^\infty r^{n-1}\vol_n(K\cap(re_n+K))dr\leq\frac{(\vol_n(K))^{n+1}}{(\vol_{n-1}(P_{e_n^\perp}(K)))^n}, $$ where $(e_i)_{i=1}^n$ denotes the canonical orthonormal basis in $\R^n$, $P_{e_n^\perp}(K)$ denotes the orthogonal projection of $K$ onto the linear hyperplane orthogonal to $e_n$, and $\vol_k$ denotes the $k$-dimensional Lebesgue measure. This inequality was proved by Gardner and Zhang and it implies Zhang's inequality. We will use our new approach to this inequality in order to prove discrete analogues of this inequality and of an equivalent version of it, where we will consider the lattice point enumerator measure instead of the Lebesgue measure, and show that from such discrete analogues we can recover the aforementioned inequality and, therefore, Zhang's inequality.