  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.
