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.