Computable Operations on Compact Subsets of Metric Spaces with Applications to Fréchet Distance and Shape Optimization
For researchers in computational geometry and shape optimization, this provides a theoretical foundation for computing with compact metric spaces, though the results are theoretical rather than practical.
The paper extends the theory of computation on real numbers and Euclidean subsets to compact metric spaces, enabling the computability of Fréchet distances between curves/loops and constrained shape optimization problems.
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.