A brief survey on lattice zonotopes

Abstract

Zonotopes are a rich and fascinating family of polytopes, with connections to many areas of mathematics. In this article we provide a brief survey of classical and recent results related to lattice zonotopes. Our emphasis is on connections to combinatorics, both in the sense of enumeration (e.g. Ehrhart theory) and combinatorial structures (e.g. graphs and permutations).Definition 2.1. Consider the polytopes, P 1 , P 2 , . . . , P m ⊂ R n . We define the Minkowski sum of the m polytopes as P 1 + P 2 + · · · + P m := {x 1 + x 2 + · · · + x m : x j ∈ P j for 1 ≤ j ≤ m} .Given v, w ∈ R n , we write [v, w] for the line segment from v to w.

Isoperimetric problems for zonotopes

“…Proof. As both sides in (14) are continuous functions of their variables, the inequality in (14) clearly holds. Similarly, the equality case in Lemma 4 holds if all 𝑥 𝑖 are positive.…”

“…being the projection bodies of convex polytopes in ℝ 𝑑 (see also [31]). They are closely related to parallelohedra, the convex polytopes whose translates fill the space (see [18,36]), and appear in lattice geometry (see, e.g., [4,28,32] or the survey [14]). Zonotopes are also investigated and often applied outside pure mathematics [1,5,6,26].…”

“…They play a central role in the theory of projections of convex polytopes, both being the projections of affine cubes (see, e.g., [37]), and, by Cauchy's projection formula [21], being the projection bodies of convex polytopes in $$\mathbb {R}^d$$ (see also [31]). They are closely related to parallelohedra, the convex polytopes whose translates fill the space (see [18, 36]), and appear in lattice geometry (see, e.g., [4, 28, 32] or the survey [14]). Zonotopes are also investigated and often applied outside pure mathematics [1, 5, 6, 26].…”