关于截面的Rogers-Shepard型不等式
Rogers-Shepard Type Inequalities for Sections
摘要：
在本文中，我们解决了以下问题：给定$\mathbb{R}^n$上的测度$μ$，是否存在一个常数$C>0$，使得对于任何$m$维子空间$H \subset \mathbb{R}^n$和任何凸体$K \subset \mathbb{R}^n$，下面的分段Rogers-Shephard型不等式成立：\[ μ((K-K) \cap H) \leq C \sup_{y \in \mathbb{R}^n} μ(K \cap (H+y))? \]我们证明了这个不等式在具有常数$C(n,m) = \binom{n+m}{m}$的径向递减密度的测度类中是肯定的。我们还证明了边际不等式的Rogers-Shephard型$\left(\frac{1}{s}\right)$ -凹，$0 \leq s < \infty$，和grammically凹函数。
英文摘要：
In this paper we address the following question: given a measure $μ$ on $\mathbb{R}^n$, does there exists a constant $C>0$ such that, for any $m$-dimensional subspace $H \subset \mathbb{R}^n$ and any convex body $K \subset \mathbb{R}^n$, the following sectional Rogers-Shephard type inequality holds: \[ μ((K-K) \cap H) \leq C \sup_{y \in \mathbb{R}^n} μ(K \cap (H+y))? \] We show that this inequality is affirmative in the class of measures with radially decreasing densities with the constant $C(n,m) = \binom{n+m}{m}$. We also prove marginal inequalities of the Rogers-Shephard type for $\left(\frac{1}{s}\right)$-concave, $0 \leq s < \infty$, and logarithmically concave functions.
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