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Valuations on Lattice Polytopes Ludwig, Monika


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 Minkowski and tensor valuations on lattice polytopes. The most important tensor valuations are the discrete moment tensor 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 $x\in{\mathbb R}^n$, and its coefficients in the Ehrhart polynomial, called Ehrhart tensors. However, there are additional examples for tensors of rank nine with the same covariance properties. (Based on joint work with K\'aroly J. B\"or\"ozcky and Laura Silverstein)

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