Closed Choice for Finite and for Convex Sets

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Abstract

We investigate choice principles in the Weihrauch lattice for finite sets on the one hand, and convex sets on the other hand. Increasing cardinality and increasing dimension both correspond to increasing Weihrauch degrees. Moreover, we demonstrate that the dimension of convex sets can be characterized by the cardinality of finite sets encodable into them. Precisely, choice from an n + 1 point set is reducible to choice from a convex set of dimension n, but not reducible to choice from a convex set of dimension n − 1.