## Unimodality problems in Ehrhart theory.(English)Zbl 1366.52014

Beveridge, Andrew (ed.) et al., Recent trends in combinatorics. Cham: Springer (ISBN 978-3-319-24296-5/hbk; 978-3-319-24298-9/ebook). The IMA Volumes in Mathematics and its Applications 159, 687-711 (2016).
This well-written survey paper gives the state of the art regarding unimodality questions of Ehrhart $$h^*$$-vectors. If $$P \subset\mathbb R^d$$ is a lattice polytope, i.e., the convex hull of finitely many points in $$\mathbb Z^d$$, then the counting function $$E(t) := \# ( tP \cap \mathbb Z^d )$$ is a polynomial in the positive integer variable $$t$$ [E. Ehrhart, C. R. Acad. Sci., Paris 254, 616–618 (1962; Zbl 0100.27601)]. When written in terms of the polynomial basis $${t \choose d}, {{t+1} \choose d}, \dots, {{t+d} \choose d}$$, the coefficients of $$E(t)$$ are nonnegative, due to a theorem of R. P. Stanley [Ann. Discrete Math. 6, 333–342 (1980; Zbl 0812.52012)]. It is natural to ask whether these coefficients – which form the $$h^*$$-vector of $$P$$ – satisfies further constraints, the most natural among which is unimodality, i.e., for some $$k$$, $h_0^* \leq \cdots \leq h_k^* \geq \cdots \geq h_d^* \, .$ Several of the known techniques to prove unimodality of a given class of $$h^*$$-vectors are algebraic in nature, due to the connection between lattice polytopes and Cohen-Macaulay semigroup algebras. The paper surveys these and many others unimodality results in this area of geometric combinatorics, and states a number of open problems.
For the entire collection see [Zbl 1348.05002].

### MSC:

 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) 05A20 Combinatorial inequalities 52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry

### Keywords:

Ehrhart theory; lattice polytopes; unimodality; log concavity

### Citations:

Zbl 0100.27601; Zbl 0812.52012
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