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Combinatorial properties of nonarchimedean convex sets

Artem Chernikov and Alex Mennen

Vol. 323 (2023), No. 1, 1–30
DOI: 10.2140/pjm.2023.323.1
Abstract

We study combinatorial properties of convex sets over arbitrary valued fields. We demonstrate analogs of some classical results for convex sets over the reals (for example, the fractional Helly theorem and Bárány’s theorem on points in many simplices), along with some additional properties not satisfied by convex sets over the reals, including finite breadth and VC dimension. These results are deduced from a simple combinatorial description of modules over the valuation ring in a spherically complete valued field.

Keywords
nonarchimedean fields, valued fields, combinatorial convexity, Helly theorem, Bárány theorem, VC dimension, breadth
Mathematical Subject Classification
Primary: 12J25, 52A01, 52A20, 52A35
Milestones
Received: 16 September 2021
Revised: 13 January 2023
Accepted: 15 March 2023
Published: 29 May 2023
Authors
Artem Chernikov
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA
United States
Alex Mennen
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA
United States

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