Banach-like metrics and metrics of compact sets

TL;DR

This paper introduces a family of geometric metrics on the space of compact sets in R^N, enabling analysis of shape properties with features like minimal geodesics and a Riemannian framework, improving regularity over Hausdorff metrics.

Contribution

The paper develops new Banach-like metrics for shape spaces that are rotation and translation invariant, with properties like geodesic minimality and a Riemannian structure, enhancing shape analysis tools.

Findings

Existence of minimal geodesics in shape space

Metrics are topologically equivalent to Hausdorff metric

Metrics exhibit better regularity and local uniqueness of geodesics

Abstract

We present and study a family of metrics on the space of compact subsets of $R^N$ (that we call ``shapes''). These metrics are ``geometric'', that is, they are independent of rotation and translation; and these metrics enjoy many interesting properties, as, for example, the existence of minimal geodesics. We view our space of shapes as a subset of Banach (or Hilbert) manifolds: so we can define a ``tangent manifold'' to shapes, and (in a very weak form) talk of a ``Riemannian Geometry'' of shapes. Some of the metrics that we propose are topologically equivalent to the Hausdorff metric; but at the same time, they are more ``regular'', since we can hope for a local uniqueness of minimal geodesics. We also study properties of the metrics obtained by isometrically identifying a generic metric space with a subset of a Banach space to obtain a rigidity result.