Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory

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On the Cheeger sets in strips and non-convex domains (2014, methodology): “We choose a point z ∈ ∂∗E ∩ Ω and, by standard estimates (see [20]), we consider a oneparameter family of diffeomorphisms equal to the identity outside a small ball Bε(z), that allow us to produce any sufficiently small volume adjustment ∆V with a change in perimeter bounded by c1∆V , for some const”
Regularity of nonlocal minimal cones in dimension 2 (2012, background): “17 in [5], where the density estimates, the compactness arguments and the monotonicity formulas for classical minimal surfaces are replaced by the ones in [2].”
Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, background): “The isoperimetric problem has been analyzed from several complementary points of view: for an overview of the more geometric aspects we refer to [55, 61, 62], for the approach via geometric measure theory see for instance [47, 51], for the connections with convex and integral geometry see [12], for ”
Minkowski Inequalities via Nonlinear Potential Theory (2019, background): “We refer the reader to [48] for a comprehensive treatment of the basic notions that we are going to recall.”