3 research outputs found
Combinatorial properties of non-archimedean convex sets
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.Comment: v.2: 27 pages; some minor corrections following referees' reports;
added a brief discussion of the other notions of convexity in valued fields
(Section 5.2) and connections to the study of abstract convexity spaces
(Section 5.3); accepted to the Pacific Journal of Mathematic