A Canonical Infinitesimally Hilbertian Structure on Locally Minkowski Spaces

The aim of this paper is to show the existence of a canonical distance d' d ′ defined on a locally Minkowski metric measure space (X, d, m) ( X , d , m ) such that: d' d ′ is equivalent to d d , (X, d', m) ( X , d ′ , m ) is infinitesimally Hilbertian. This new regularity assumption on (X, d, m) ( X , d , m ) essentially forces the structure to be locally similar to a Minkowski space and defines a class of metric measure structures which includes all the reversible Finsler manifolds and it is actually strictly larger. The required distance d' d ′ will be the intrinsic distance dKS d KS associated to the so-called Korevaar-Schoen energy , which is proven to be a quadratic form. In particular, we show that the Cheeger energy associated to the metric measure space (X, dKS, m) ( X , d KS , m ) is in fact the Korevaar-Schoen energy.