Combinatorial properties of nonarchimedean convex sets
نویسندگان
چکیده
We study combinatorial properties of convex sets over arbitrary valued fields. demonstrate analogs some classical results for the reals (e.g. fractional Helly theorem and B\'ar\'any's on points in many simplices), along with additional not satisfied by reals, including finite breadth VC-dimension. These are deduced from a simple description modules valuation ring spherically complete field.
منابع مشابه
Combinatorial properties of convex sets Lecture 5 — 23 / 01 / 2015
i=1 αif(vi), where x = ∑d+1 i=1 αivi ∈ ∂X and for i = 1, ..., d+1 αi ≥ 0 and ∑d+1 i=1 α = 1. For every i = 1, ..., d+1, vi ∈ {−ei, ei}. By the Borsuk-Ulam theorem, there exists x ∈ ∂X s.t. f(x) = f(−x). Hence, there exist α1, ..., αd+1, v1, ..., vd+1 s.t. ∑d+1 i=1 αif(vi) = ∑d+1 i=1 αif(−vi). By definition the two simplices conv(f(v1), ..., f(vd+1)) and conv(f(−v1), ..., f(−vd+1)) cover each co...
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ژورنال
عنوان ژورنال: Pacific Journal of Mathematics
سال: 2023
ISSN: ['1945-5844', '0030-8730']
DOI: https://doi.org/10.2140/pjm.2023.323.1