Isoperimetric-type inequalities

Abstract

It is a fact of experience that several inequalities with isoperimetric content can be retrieved by considering the above-tangent formulation of displacement convexity. Here is a possible heuristic explanation for this phenomenon. Assume, for the sake of the discussion, that the initial measure is the normalized indicator function of some set A. Think of the functional U_v as the internal energy of some fluid that is initially confined in A. In a displacement interpolation, some of the mass of the fluid will have to flow out of A, leading to a variation of the energy (typically, more space available means less density and less energy). The decrease of energy at initial time is related to the amount of mass that is able to flow out of A at initial time, and that in turn is related to the surface of A (a small surface leads to a small variation, because not much of the fluid can escape). So by controlling the decrease of energy, one should eventually gain control of the surface of A.