Combinatorial properties of nonarchimedean convex sets

Artem Chernikov, Alex Mennen

Type: Article

Publication Date: 2023-05-29

Citations: 2

DOI: https://doi.org/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.

Locations

Pacific Journal of Mathematics - View - PDF
eScholarship (California Digital Library) - View - PDF
arXiv (Cornell University) - View - PDF

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+	Vapnik-Chervonenkis density in some theories without the independence property, I	2015	Matthias Aschenbrenner Alf Dolich Deirdre Haskell Dugald Macpherson Sergei Starchenko
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