A Metric Approach to Elastic Reformations

We study a variational framework to compare shapes, modeled as Radon measures on R^N, in order to quantify how they differ from isometric copies. To this purpose we discuss some notions of weak deformations termed reformations as well as integral functionals having some kind of isometries as minimizers. The approach pursued is based on the notion of pointwise Lipschitz constant leading to a space metric framework. In particular, to compare general shapes, we study this reformation problem by using the notion of transport plan and of Wasserstein distances as in optimal mass transportation theory.