Banach-like metrics and metrics of compact sets

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.

💡 Research Summary

The paper introduces a family of metrics on the collection of compact subsets of ℝⁿ—referred to as “shapes”—by embedding these sets into Banach (or Hilbert) spaces and measuring distances with the ambient norm. The authors begin by representing each compact set S through its indicator function χ_S or, more smoothly, through its distance function dist_S(x)=inf_{y∈S}‖x−y‖. By mapping S to a function in L²(ℝⁿ) (or in a Sobolev space H¹(ℝⁿ)), the distance between two shapes S₁ and S₂ is defined as the norm of the difference of the corresponding functions: \