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