Workshop

Lattice polytopes are convex hulls of finitely many points with integer coordinates in ${\mathbb R}^n$. A function $Z$ from a family ${\cal F}$ of subsets of ${\mathbb R}^n$ with values in an abelian group is a valuation if $$ Z(P)+Z(Q)=Z(P\cup Q)+Z(P\cap Q) $$ whenever $P,Q,P\cup Q,P\cap Q\in{\cal F}$ and $Z(\emptyset)=0$. The classification of real-valued invariant valuations on lattice polytopes by Betke \& Kneser is classical (and will be recalled). It establishes a characterization of the coefficients of the Ehrhart polynomial. Building on this, classification results are established for vector, matrix, and general tensor valuations on lattice polytopes. The most important tensor valuations are the discrete moment tensors of rank $r$, $$ L^r(P)=\frac1{r!}\sum_{x\in P\cap{\mathbb Z}^n}x^r, $$ where $x^r$ denotes the $r$-fold symmetric tensor product of the integer point $x\in{\mathbb Z}^n$, and its coefficients in the Ehrhart tensor polynomial, called Ehrhart tensors. However, it is shown that there are additional examples for tensors of rank nine with the same covariance properties. For tensors of rank up to eight, a complete classification is established

Many important features of the geometry of high-dimensional spaces can be seen as consequences of concentration of measure on the orthogonal group.  I will illustrate this phenomenon with three examples: the Johnson-Lindenstrauss lemma, Dvoretzky's theorem, and a measure theoretic analog of the latter.  The talk will be elementary and assume no prior knowledge of measure concentration or any of the results to be discussed.

We prove that in every n-dimensional there exists a constant c=c(n)>0 so that in the c(n)-neighborhood of a ball, the only convex body with uniform cone volume measure is the ball. The goal of the talk will be to give an insight into some analytic aspects of the Log-Brunn-Minkowski and Log-Minkowski conjectures made by Boroczky, Lutwak, Yang and Zhang. This talk is based on the joint papers with Colesanti, Marsiglietti and Colesanti.