Luca Granieri

A Metric Approach to Elastic Reformations

We study a variational framework to compare shapes, modeled as Radon measures on

Inverse Function Theorems and Jacobians over Metric Spaces

{\mathbb{R}}^{N}

Remarks on the stability of mass minimizing currents in the Monge-Kantorovich problem

, 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 matric space framework. In particular, to compare general shapes, we study this reformation problem by using the notion of transport plan and Wasserstein distances as in optimal mass transportation theory.

On a distance representation of Kantorovich potentials

Abstract We present inversion results for Lipschitz maps f : Ω ⊂ ℝN → (Y, d) and stability of inversion for uniformly convergent sequences. These results are based on the Area Formula and on the l.s.c. of metric Jacobians.

Minimal measures, one-dimensional currents and the Monge-Kantorovich problem

Abstract The Monge-Kantorovich problem is equivalent to the problem of finding 1-currents with fixed boundary and minimal mass. We address the question of the stability for the mass minimizing currents. In particular, we state a Γ-convergence result. We provide proofs relying just on basic properties of currents and on the notion of flat norm.

On action minimizing measures for the Monge-Kantorovich problem

Abstract We address the question of how to represent Kantorovich potentials in the mass transportation (or Monge–Kantorovich) problem as a signed distance function from a closed set. We discuss geometric conditions on the supports of the measure f + and f − in the Monge–Kantorovich problem which ensure such a representation. Finally, we obtain, as a by-product, the continuous differentiability of the potential on the transport set.