Computable Operations on Compact Subsets of Metric Spaces with Applications to Fr\'echet Distance and Shape Optimization

TL;DR

This paper extends the theory of computation to compact metric spaces, enabling the computability of Fréchet distances and shape optimization problems in more general settings.

Contribution

It generalizes computational and optimization frameworks from Euclidean spaces to arbitrary compact metric spaces, preserving structural properties like Cartesian closure.

Findings

Computability of Fréchet distances between curves and loops.

Computability of constrained and shape optimization problems.

Structural preservation of computational properties in general metric spaces.

Abstract

We extend the Theory of Computation on real numbers, continuous real functions, and bounded closed Euclidean subsets, to compact metric spaces $(X,d)$: thereby generically including computational and optimization problems over higher types, such as the compact 'hyper' spaces of (i) nonempty closed subsets of $X$ w.r.t. Hausdorff metric, and of (ii) equicontinuous functions on $X$. The thus obtained Cartesian closure is shown to exhibit the same structural properties as in the Euclidean case, particularly regarding function pre/image. This allows us to assert the computability of (iii) Fr\'echet Distances between curves and between loops, as well as of (iv) constrained/Shape Optimization.