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BIRS Workshop Lecture Videos

A reverse Rogers-Shephard ineqaulity for log-concave functions Alonso, David


The classical Rogers-Shephard inequality provides an upper bound of the volume of the difference body of an $n$-dimensional convex body $K$ and, more generally, it states that for any pair of convex bodies $K,L$ and any $x_0\in\R^n$ $$ |K\cap (x_0+L)||K-L|\leq{2n\choose n}|K||L|, $$ with equality if and only if $K=x_0+L$ is a simplex. A reverse inequality was given by Milman and Pajor, showing that for any pair of convex bodies with the same barycenter $$ |K||L|\leq|K-L||K\cap L|. $$ In this talk we will extend these inequalities to the more general setting of log-concave functions, showing that for any pair of integrable log-concave functions $f,g$, if $f\star g$ denotes their Asplund product and $f*g$ their convolution, we have that $$ \Vert f*g\Vert_\infty\Vert f\star g\Vert_1\leq {2n\choose n}\Vert f\Vert_\infty\Vert g\Vert_\infty\Vert f\Vert_1\Vert g\Vert_1, $$ with equality if and only if $\frac{f(x)}{\Vert f\Vert_\infty}=\frac{g(-x)}{\Vert g\Vert_\infty}$ is the characteristic function of a simplex and if $f$ and $g$ have opposite barycenters and attain their maximums at 0 $$ \Vert f\Vert_\infty\Vert g\Vert_\infty\Vert f\Vert_1\Vert g\Vert_1\leq e^{1+\textrm{Ent}\left(\frac{f}{\Vert f\Vert_\infty}\right)+\textrm{Ent}\left(\frac{g}{\Vert g\Vert_\infty}\right)} f*g(0)\Vert f\star g\Vert_1, $$ improving the value of the constant in some particular cases.

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