Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Eric Shen
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7201868256