The Zaporozhets-Tarasov Inequality for an Arbitrary Planar Convex Body

[2608.14833] The Zaporozhets-Tarasov Inequality for an Arbitrary Planar Convex Body Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Metric Geometry arXiv:2608.14833 (math) [Submitted on 14 Aug 2026] Title: The Zaporozhets-Tarasov Inequality for an Arbitrary Planar Convex Body Authors: Maksim Kukushkin View a PDF of the paper titled The Zaporozhets-Tarasov Inequality for an Arbitrary Planar Convex Body, by Maksim Kukushkin View PDF HTML (experimental) Abstract: We prove that the mean distance between two independent uniformly distributed points in an arbitrary planar convex body is strictly smaller than the mean distance between two independent uniformly distributed points on its boundary. Equality is impossible, although the difference between these two mean distances tends to zero along a sequence of thin rectangles. The proof reduces the problem to a one-dimensional comparison of two Gini mean differences. The main new ingredient is a moment lemma for a random pair $(\varepsilon,R)\in\{-1,1\}\times[0,1]$. The only finite algebraic part of the proof is given by exact rational certificates in the Bernstein basis; the verification script and all 6492 coefficients are available in the supplementary materials. Comments: 35 pages, 3 figures. Russian manuscript completed and privately deposited on GitHub on July 26, 2026: this https URL . Repository public since August 13, 2026. English translation; mathematical text unchanged except for an Author's note Subjects: Metric Geometry (math.MG) ; Probability (math.PR) MSC classes: 52A40 (Primary), 60D05 (Secondary) Cite as: arXiv:2608.14833 [math.MG] (or arXiv:2608.14833v1 [math.MG] for this version) https://doi.org/10.48550/arXiv.2608.14833 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Maksim Kukushkin [ view email ] [v1] Fri, 14 Aug 2026 19:10:21 UTC (64 KB) Full-text links: Access Paper: View a PDF of the paper titled The Zaporozhets-Tarasov Inequality for an Arbitrary Planar Convex Body, by Maksim Kukushkin View PDF HTML (experimental) TeX Source view license Ancillary-file links: Ancillary files ( details ) : README.txt moment_bernstein_coefficients.txt moment_certificates.log moment_certificates_stdlib.log requirements.txt symbolic_identities.log verify_certificates_stdlib.py verify_moment_lemma_certificates.py verify_symbolic_identities.py (4 additional files not shown) 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

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