可计算操作在度量空间紧子集上的扩展及其在弗雷歇距离和形状优化中的应用
Computable Operations on Compact Subsets of Metric Spaces with Applications to Fréchet Distance and Shape Optimization
AI总结：
本文扩展了实数、连续实函数和有界闭欧几里得子集的计算理论到紧度量空间，探讨了高阶类型中的计算与优化问题，包括基于豪斯多夫度量的非空闭子集和等连续函数的紧超空间，并证明了弗雷歇距离和约束形状优化的可计算性。
摘要：
我们扩展了实数、连续实函数和有界闭欧几里得子集的计算理论到紧度量空间（X,d）：从而普遍包括高阶类型中的计算和优化问题，例如基于豪斯多夫度量的非空闭子集的紧超空间，以及X上的等连续函数。由此获得的笛卡尔闭包被证明具有与欧几里得情况相同的结构性质，特别是关于函数前像。这使得我们可以断言（iii）曲线和环之间的弗雷歇距离以及（iv）约束/形状优化的可计算性。
英文摘要：
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.
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