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A continuous family of marked poset polytopes. (English) Zbl 1445.52009
To a finite poset \((P, \preceq)\) Stanley [R. P. Stanley, Discrete Comput. Geom. 1, 9–23 (1986; Zbl 0595.52008)] famously associated two lattice polytopes: the order polytope \[ O_{P, \preceq} := \left\{ x \in [0,1]^P :\, p \preceq q \ \Rightarrow \ x_p \leq x_q \right\} \] and the chain polytope \[ C_{P, \preceq} := \left\{ x \in \mathbb{R}_{\ge 0}^P :\, p_1 \prec \cdots \prec p_m \text { maximal chain in } P \ \Rightarrow \ x_{p_1} + \cdots + x_{p_m} \le 1 \right\} . \] These polytopes have seen much research activity and were generalized to marked poset polytopes [F. Ardila et al., J. Comb. Theory, Ser. A 118, No. 8, 2454–2462 (2011; Zbl 1234.52009)], where now some of the poset elements are labelled with fixed integers. The paper under review defines and studies a continuous family of polytopes with marked order and chain polytope at extreme ends. The authors study the Ehrhart theory (lattice-point counting formulas) and the combinatorial type of these polytopes, using tools from discrete and tropical geometry.

52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
06A07 Combinatorics of partially ordered sets
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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