关于平均距离的扎波罗热茨-塔拉索夫猜想

中文摘要 AI 辅助

对于凸体$K \subset \mathbb R^d$，令$\Delta(K)$为$K$中两个独立均匀点之间的期望距离，$\theta(K)$为$\partial K$上归一化曲面测度对应的期望。扎波罗热茨-塔拉索夫猜想断言$\Delta(K) \le \theta(K)$。我们证明了$d=2$时该猜想成立；此外，给出了$\mathbb R^3$中一个六顶点凸多面体，其反向严格不等式成立，且通过与线段取乘积得到所有$d \ge 3$维的反例。

英文摘要

For a convex body $K \subset \mathbb R^d$ let $Δ(K)$ be the expected distance between two independent uniform points of $K$, and let $θ(K)$ be the corresponding expectation for normalized surface measure on $\partial K$. The Zaporozhets-Tarasov conjecture asserts $Δ(K) \le θ(K)$. We prove this conjecture in case $d=2$. In addition, we give a six-vertex convex polytope in $\mathbb R^3$ for which the reverse strict inequality holds, and obtain counterexamples in every dimension $d \ge 3$ by taking products with segments. Finally, we show that $θ(K)> \frac{\operatorname{per}K}{6}$ for every planar convex body.

发表机构

• Moscow State University（莫斯科国立大学）
• Institute for Numerical Mathematics, Russian Academy of Sciences（俄罗斯科学院计算数学研究所）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

备注

v2 Added Section 5, a note about works of Kukushkin and Lotnikov, Declaration of AI use and Acknowledgments. v3 made proof of Theorem 1.2 shorter, proved analog of the ZT conjecture for the second moment (Corr. 3.3) Proved (strict form) $θ(K)> \frac{\operatorname{per}K}{6}$, simplified section 5, in particular shortened the proof of Lemma 5.4. Corrected the title of reference [3]