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PreprintCC BY 4.0arXiv

The Zaporozhets-Tarasov Conjecture on Mean Distances

  • Eric Shen
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Abstract

For a convex body K ⊂ ℝd 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 ∂ K. The Zaporozhets-Tarasov conjecture asserts Δ(K) ≤ θ(K). We prove this conjecture in case d=2. In addition, we give a six-vertex convex polytope in ℝ3 for which the reverse strict inequality holds, and obtain counterexamples in every dimension d ≥ 3 by taking products with segments. Finally, we show that θ(K)> perK/6 for every planar convex body.

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Cite

BibTeXRIS
BibTeX
@misc{shen2026zaporozhetstarasov,
  title = {{The Zaporozhets-Tarasov Conjecture on Mean Distances}},
  author = {Eric Shen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.06470},
  eprint = {2608.06470},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.06470},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Zaporozhets-Tarasov Conjecture on Mean Distances
AU  - Eric Shen
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2608.06470
UR  - https://arxiv.org/abs/2608.06470
AB  - For a convex body K ⊂ 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 ∂ K. The Zaporozhets-Tarasov conjecture asserts Δ(K) ≤ θ(K). We prove this conjecture in case d=2. In addition, we give a six-vertex convex polytope in ℝ^3 for which the reverse strict inequality holds, and obtain counterexamples in every dimension d ≥ 3 by taking products with segments. Finally, we show that θ(K)> perK/6 for every planar convex body.
ER  -