On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube

[2608.14909] On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Metric Geometry arXiv:2608.14909 (math) [Submitted on 14 Aug 2026] Title: On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube Authors: Gennadiy Averkov , Katherina von Dichter , Ivan Soprunov View a PDF of the paper titled On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube, by Gennadiy Averkov and 2 other authors View PDF HTML (experimental) Abstract: We prove a log-submodularity-type inequality for zonoids in $\mathbb{R}^4$, extending the three-dimensional result of Fradelizi, Madiman, Meyer, and Zvavitch. More generally, we conjecture a log-submodularity-type inequality for zonoids in arbitrary dimension. This inequality admits several equivalent formulations in terms of volumes of coordinate projections as well as in terms of mixed volumes, thereby unifying several geometric perspectives. We reduce the conjectured inequality to a polynomial inequality whose variables are associated with the vertices of a hypercube and whose coefficients encode the volumes of 0/1 simplices. This reduction reveals unexpected connections between mixed volumes of zonoids, matroid theory, and real algebraic geometry. Comments: 24 pages Subjects: Metric Geometry (math.MG) ; Combinatorics (math.CO) MSC classes: Primary 52A39, Secondary 52A40, 14M15, 14Q30 Cite as: arXiv:2608.14909 [math.MG] (or arXiv:2608.14909v1 [math.MG] for this version) https://doi.org/10.48550/arXiv.2608.14909 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Ivan Soprunov [ view email ] [v1] Fri, 14 Aug 2026 21:47:15 UTC (41 KB) Full-text links: Access Paper: View a PDF of the paper titled On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube, by Gennadiy Averkov and 2 other authors 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.CO 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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