We prove metric differentiation for differentiability spaces in the sense of Cheeger [10, 14,
27]. As corollarieswe give a new proof of one of the main results of [14], a proof that the Lip …

We characterize complete RNP-differentiability spaces as those spaces which are rectifiable
in terms of doubling metric measure spaces satisfying some local (1, p)-Poincaré …

The goal of this paper is to continue the study of the relation between the Poincaré inequality
and the lower bounds of Minkowski content of separating sets, initiated in our previous work …

There has been considerable effort to better understand the generalization capabilities of
deep neural networks both as a means to unlock a theoretical understanding of their …

Here we show existence of many subsets of Euclidean spaces that, despite having empty
interior, still support Poincaré inequalities with respect to the restricted Lebesgue measure …

This paper introduces and studies the analogue of the notion of Lipschitz differentiability
space by Cheeger, using Carnot groups and Pansu derivatives as models. We call such …

We construct an isometric embedding from Gigli's abstract tangent module into the concrete
tangent module of a space admitting a (weak) Lipschitz differentiable structure, and give two …

In this work we give a sufficient condition under which the global Poincar\'{e} inequality on
Carnot groups holds true for a large family of probability measures absolutely continuous …

The $ p $-modulus of curves, test plans, upper gradients, charts, differentials,
approximations in energy and density of directions are all concepts associated to the theory …

It is a major problem in analysis on metric spaces to understand when a metric space is
quasisymmetric to a space with strong analytic structure, a so-called Loewner space. A …