Banach-like metrics and metrics of compact sets

We present and study a family of metrics on the space of compact subsets of R (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 general properties of the metrics obtained by isometrically identifying a generic metric space with a subset of a Banach space and we obtain a rigidity result. AMS 2000 Mathematics Subject Classification 54E35 58B20 58D15