Abstract
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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.
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Keywords
nonarchimedean fields, valued fields, combinatorial
convexity, Helly theorem, Bárány theorem, VC dimension,
breadth
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Mathematical Subject Classification
Primary: 12J25, 52A01, 52A20, 52A35
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Milestones
Received: 16 September 2021
Revised: 13 January 2023
Accepted: 15 March 2023
Published: 29 May 2023
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© 2023 The Author(s), under
exclusive license to MSP (Mathematical Sciences Publishers).
Distributed under the Creative Commons
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