Calculus of Variations and Geometric Measure Theory

A. C. G. Mennucci - A. Duci

Banach-like distances and metrics spaces of compact sets

created by mennucci on 06 Nov 2007
modified on 07 Jan 2015


Published Paper

Inserted: 6 nov 2007
Last Updated: 7 jan 2015

Journal: SIAM Journal on Imaging Sciences
Volume: 8
Number: 1
Pages: 19-66
Year: 2015
Doi: 10.1137/140972512


In the first part we study general properties of the metrics obtained by isometrically identifying a generic metric space with a subset of a Banach space; we obtain a rigidity result. We then discuss the Hausdorff distance, proposing some less--known but important results: a closed--form formula for geodesics; generically two compact sets are connected by a continuum of geodesics. In the second part we present and study a family of distances on the space of compact subsets of $R^N$ (that we call ``shapes''). These distances are ``geometric'', that is, they are independent of rotation and translation; and the resulting metric spaces enjoy many interesting properties, as, for example, the existence of geodesics. We view our metric space of shapes as a subset of Banach (or Hilbert) spaces: 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 distance.

Keywords: Hausdorff distance, Shape, isometric embedding, 54E35, 58B20, 58D15