Geometry Seminar – Jesús Yepes

Event Details

Title: Una caracterización del volumen mediante la desigualdad de Brunn-Minkowski

Speaker: Jesús Yepes

The Brunn-Minkowski inequality can be summarized by stating that the volume, i.e., the Lebesgue measure in R^n, is (1/n)-concave. More precisely, we have

(1) vol((1−λ)A+λB)^1/n≥(1−λ)vol(A)^1/n+λvol(B)^1/n

for all measurable sets A,B so that (1−λ)A+λB is also measurable. It is easy to see that (1) is also true if we exchange 1/n by an arbitrary p≤1/n. When working with absolutely continuous measures dμ(x)=f(x)dx associated to densities f with some convexity assumptions, one can also obtain the following Brunn-Minkowski inequality

(2) μ((1−λ)A+λB)^p≥(1−λ)μ(A)^p+λμ(B)^p

for any pair of measurable sets A,B with μ(A)μ(B)>0 and such that (1−λ)A+λB is also measurable, where p≤1/n is associated to the ``type of convexity'' of f. The convexity conditions of such density functions f allow us to understand whether the volume should be the sole measure satisfying the latter inequality or not. Thus, in this talk we will discuss whether, for a given measure on R^n (not necessarily absolutely continuous), having an inequality like (2) for a certain (`small') subfamily of sets in R^n implies that the measure is (up to a constant) the volume itself. To this respect, we will point out that the constraints 1/n≥p>0, when the support of the measure is the whole R^n, and p=1/n, when it is an arbitrary open convex set, are both necessary in order to get such a characterization

3 June 2016, 11:30, 1st floor Seminar room, IEMath-GR

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