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We prove that on an arbitrary metric measure space a countable collection of test plans is sufficient to recover all $rm BV$ functions and their total variation measures. In the setting of non-branching ${sf CD}(K,N)$ spaces (with finite reference measure), we can additionally require these test plans to be concentrated on geodesics.
We study $L^p$ Besov critical exponents and isoperimetric and Sobolev inequalities associated with fractional Laplacians on metric measure spaces. The main tool is the theory of heat semigroup based Besov classes in Dirichlet spaces that was introduced by the authors in previous works.
We discuss a new pseudometric on the space of all norms on a finite-dimensional vector space (or free module) $mathbb{F}^k$, with $mathbb{F}$ the real, complex, or quaternion numbers. This metric arises from the Lipschitz-equivalence of all norms on
We show that every finite-dimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.
In a seminal paper Volumen und Oberflache (1903), Minkowski introduced the basic notion of mixed volumes and the corresponding inequalities that lie at the heart of convex geometry. The fundamental importance of characterizing the extremals of these
In this work we study the issue of geodesic extendibility on complete and locally compact metric length spaces. We focus on the geometric structure of the space $(Sigma (X),d_H)$ of compact balls endowed with the Hausdorff distance and give an explic