The Isoperimetric Problem

By Viktor Blåsjö. The American mathematical monthly, 2005

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Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry (2004, background): “Theorem 10 in [10] asserts that among measures having the same concave isoperimetric behaviour, the even logconcave one minimizes the isoperimetric profile for the product measures, see also [49].”
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 ”
Free boundary minimal surfaces of unbounded genus (2016, background): “(Almgren [Al], Bokowski-Sperner [BS], Ros [R]) The isoperimetric surfaces in B are hyperplanes through the origin or spherical caps meeting ∂B orthogonally.”
Minimal surfaces in $$S^3$$: a survey of recent results (2013, neutral): “By a theorem of Ritoré and Ros, any region in S3 which has volume 12 vol(S 3) and is invariant under antipodal symmetry has boundary area at least 2π2 (see [44] or [47], Corollary 5).”