Mean distance between points inside and on the boundary of a convex body

[2608.14848] Mean distance between points inside and on the boundary of a convex body Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Metric Geometry arXiv:2608.14848 (math) [Submitted on 14 Aug 2026] Title: Mean distance between points inside and on the boundary of a convex body Authors: A. S. Lotnikov View a PDF of the paper titled Mean distance between points inside and on the boundary of a convex body, by A. S. Lotnikov View PDF HTML (experimental) Abstract: The Zaporozhets--Tarasov conjecture is considered, which states that the mean distance between two random points inside a convex body does not exceed the mean distance between two random points on its boundary. For centrally symmetric bodies the question is settled completely: the conjecture is proved for planar bodies and disproved in every dimension at least three, for all moments of the distance; an explicit family of counterexamples is constructed together with a dimension-lifting construction. It is also shown that for sufficiently high moments an analogous inequality holds for any planar convex body. For circumscribed bodies exact relations between the mean distances are obtained --- analogues of Kingman's formula derived via the projective distributions introduced here; moreover, the conjecture is proved for triangles in a stronger per-projection form. Comments: English translation of the author's this http URL . thesis (St. Petersburg State University, defended June 15, 2026; advisor D. Zaporozhets). Apart from the Note added, the text is unchanged Subjects: Metric Geometry (math.MG) ; Probability (math.PR) Cite as: arXiv:2608.14848 [math.MG] (or arXiv:2608.14848v1 [math.MG] for this version) https://doi.org/10.48550/arXiv.2608.14848 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Alexey Lotnikov [ view email ] [v1] Fri, 14 Aug 2026 19:43:02 UTC (195 KB) Full-text links: Access Paper: View a PDF of the paper titled Mean distance between points inside and on the boundary of a convex body, by A. S. Lotnikov View PDF HTML (experimental) TeX Source view license Current browse context: math.MG < prev | next > new | recent | 2026-08 Change to browse by: math math.PR References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

Related documents

• Sharp Bohr and Bohr-Rogosinski inequalities involving area measure for close-to-convex harmonic mappings#985939
• The lower mean curvature bound in Gromov's mean-of-the-mean-curvature conjecture#821155
• An Inequality in Metric Spaces#190811
• Generic uniqueness of area minimizing disks for extreme curves#954490
• Outward bound#66361