AI 中文总结
研究凸体内部及边界上随机点，证明在特定条件下\(I\)在凸序上被\(B\)支配，得出对于凸函数\(\varphi\)的期望不等式，还得到随机弦矩及随机单纯形平均体积的不等式，证实相关猜想。
AI 中文摘要
设\(K\)为\(\mathbb{R}^d\)中的凸体，\(I\)和\(B\)分别为在\(K\)内部及其边界上均匀分布的随机点。证明了若\(d = 2\)且\(\mathbb{E}I = \mathbb{E}B\)，或者\(K\)是内切球中心与\(\mathbb{E}I = \mathbb{E}B\)重合的外切多面体，则\(I\)在凸序上被\(B\)支配。由此可得，对于每个变量都为凸函数的\(\varphi\)，期望\(\mathbb{E}\varphi(I_1,\dots,I_k)\)不超过\(\mathbb{E}\varphi(B_1,\dots,B_k)\)。特别地，得到随机弦矩之间的不等式\(\mathbb{E}|I_1 - I_2|^p \leqslant \mathbb{E}|B_1 - B_2|^p\)对所有\(p \geqslant 1\)成立，证实了特定类凸体的扎波罗热茨 - 塔拉索夫猜想，并扩展到随机单纯形平均体积的不等式。
英文摘要
Let $K$ be a convex body in $\mathbb{R}^d$, and let $I$ and $B$ be random points uniformly distributed inside $K$ and on its boundary, respectively. We prove that if $d=2$ and $\mathbb E I = \mathbb E B$, or if $K$ is a circumscribed polytope with the center of the inscribed sphere coinciding with $\mathbb E I = \mathbb E B$, then $I$ is dominated by $B$ in the convex order. As a consequence, for any function $φ$ convex in each argument, the expectation $\mathbb E φ(I_1,\dots,I_k)$ does not exceed $\mathbb E φ(B_1,\dots,B_k)$. This yields, in particular, an inequality between the moments of random chords $\mathbb E |I_1 - I_2|^p \leqslant \mathbb E |B_1 - B_2|^p$ for all $p \geqslant 1$, confirming the Zaporozhets--Tarasov conjecture for the indicated class of bodies, and extends to inequalities for mean volumes of random simplices.
