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The aim of this paper is to show the existence of a canonical distance$\mathsf d'$ defined on a locally Minkowski metric measure space $(\mathsfX,\mathsf d,\mathfrak m)$ such that:  i) $\mathsf d'$ is equivalent to $\mathsf d$,  ii) $(\mathsf X, \mathsf d', \mathfrak m)$ is infinitesimally Hilbertian.  This new regularity assumption on $(\mathsf X, \mathsf d,\mathfrak m)$essentially forces the structure to be locally similar to a Minkowski space anddefines a class of metric measure structures which includes all the Finslermanifolds, and it is actually strictly larger. The required distance $\mathsfd'$ will be the intrinsic distance $\mathsf d_\mathsf{KS}$ associated to theso-called Korevaar-Schoen energy, which is proven to be a quadratic form. Inparticular, we show that the Cheeger energy associated to the metric measurespace $(\mathsf X, \mathsf d_\mathsf{KS}, \mathfrak m)$ is in fact theKorevaar-Schoen energy.

A canonical infinitesimally Hilbertian structure on locally Minkowski spaces