Combinatorial properties of non-archimedean convex sets

TL;DR

This paper explores the combinatorial characteristics of convex sets over arbitrary valued fields, extending classical real convex set results and introducing new properties like finite breadth and VC-dimension.

Contribution

It provides the first combinatorial framework for convex sets over valued fields, including analogs of classical theorems and novel properties not seen over the reals.

Findings

Analogues of fractional Helly theorem and Baryany's theorem for valued fields

Convex sets exhibit finite breadth and VC-dimension over valued fields

A simple combinatorial description of modules over valuation rings

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 (e.g. the fractional Helly theorem and B\'ar\'any'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.