| Summary: | 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. Furthermore we consider
searching for zeros of continuous functions. We provide an algorithm producing
3n real numbers containing all zeros of a continuous function with up to n
local minima. This demonstrates that having finitely many zeros is a strictly
weaker condition than having finitely many local extrema. We can prove 3n to be
optimal.
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