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Title:
Tensor valuations on lattice polytopes
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Abstract:
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
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