The Zaporozhets-Tarasov Conjecture on Mean Distances

Abstract

For a convex body $K\subset {\mathbb{R}}^d$ let $\Delta (K)$ be the expected distance between two independent uniform points of $K$, and let $\theta (K)$ be the corresponding expectation for normalized surface measure on $\partial K$. The Zaporozhets-Tarasov conjecture asserts $\Delta (K)\le \theta (K)$. 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 $\theta (K)>\frac{\mathrm{per}K}{6}$ for every planar convex body.

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) $\theta (K)>\frac{\mathrm{per}K}{6}$, simplified section 5, in particular shortened the proof of Lemma 5.4. Corrected the title of reference [3]