Tensor valuations on lattice polytopes
Connections for Women: geometry and probability in high dimensions August 17, 2017  August 18, 2017
Location: MSRI: Simons Auditorium
Valuation
lattice polytope
Ehrhart polynomial
tensor
1Ludwig
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 realvalued 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
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