Finite choice, convex choice and finding roots

TL;DR

This paper explores the complexity of choice principles in the Weihrauch lattice for finite and convex sets, establishing relationships between set size, convex set dimension, and zero-finding algorithms for continuous functions.

Contribution

It characterizes the Weihrauch degrees of choice principles for finite and convex sets and introduces an optimal algorithm for zero-finding with specific bounds.

Findings

Choice from an n+1 point set reduces to choice from an n-dimensional convex set.

Zero-finding algorithm produces 3n real numbers containing all zeros for functions with up to n local minima.

Having finitely many zeros is weaker than having finitely many local extrema, with proven optimal bounds.

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. 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.