Title: Computable Operations on Compact Subsets of Metric Spaces with Applications to Fréchet Distance and Shape Optimization,
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) Fréchet
Distances between curves and between loops, as well as of (iv)
constrained/Shape Optimization.