[Paper Review] Cayley sums and Minkowski sums of lattice polytopes

Akiyoshi Tsuchiya|arXiv (Cornell University)|Apr 27, 2018

Commutative Algebra and Its Applications14 references3 citations

TL;DR

This paper establishes a characterization of when Cayley sums of lattice polytopes have the integer decomposition property (IDP) via their Minkowski sums, proving that dilated Cayley sums are IDP if and only if the original polytopes are IDP and their scaled tuples satisfy the IDP condition. It further shows that if each polytope is dilated by a factor of at least dim(P)+1, the resulting Cayley sum is level of index m, while the Minkowski sum is level of index 1, resolving a conjecture positively for the level property but negatively for the IDP property under weaker dilation conditions.

ABSTRACT

In this paper, we discuss the integer decomposition property for Cayley sums and Minkowski sums of lattice polytopes. In fact, we characterize when Cayley sums have the integer decomposition property in terms of Minkowski sums. Moreover, by using this characterization, we consider when Cayley sums and Minkowski sums of $2$-convex-normal lattice polytopes have the integer decomposition property. Finally, we also discuss the level property for Minkowski sums and Cayley sums.

Motivation & Objective

• To characterize when Cayley sums of lattice polytopes have the integer decomposition property (IDP) using Minkowski sum conditions.
• To investigate whether dilated Cayley sums and Minkowski sums of 2-convex-normal or 2-convex-level polytopes inherit the IDP or level property.
• To resolve Question 1.5 regarding the minimal dilation required for IDP and level properties in Cayley sums.
• To determine optimal dilation thresholds for the level property in Cayley and Minkowski sums of lattice polytopes.

Proposed method

• The paper introduces the concept of 2-convex-normal and 2-convex-level lattice polytopes to analyze the IDP and level properties under dilation.
• It establishes a key equivalence: a Cayley sum is IDP if and only if each constituent polytope is IDP and all nonnegative integer combinations of their dilations satisfy the IDP condition.
• The proof uses barycentric coordinates and lattice point decomposition in the Cayley sum construction, particularly analyzing interior lattice points in dilated polytopes.
• It applies results from Higashitani (2016) on Minkowski sums of dilated polytopes to derive conditions for levelness.
• The paper constructs counterexamples to show that the IDP condition does not hold for Cayley sums when dilation is only dim(P), even if each polytope is IDP.
• It proves that for any lattice polytope P with dim(P) = d, the dilation nP is 2-convex-level if n ≥ d+1, which is used to establish the level property of Cayley sums.

Experimental results

Research questions

• RQ1 Under what conditions is the Cayley sum of lattice polytopes IDP, and how is this related to the IDP of their Minkowski sums?
• RQ2 Is the Cayley sum of dilated polytopes IDP when each dilation factor is at least dim(P_i)?
• RQ3 Is the Cayley sum of dilated polytopes level of index m when each dilation factor is at least dim(P_i)+1?
• RQ4 What is the minimal dilation required for a Cayley sum to be level, and is this bound optimal?
• RQ5 Can the IDP and level properties of Minkowski sums and Cayley sums be characterized via 2-convex-normal or 2-convex-level polytopes?

Key findings

• The Cayley sum P₁*⋯*Pₘ is IDP if and only if each Pᵢ is IDP and all tuples (a₁P₁,…,aₘPₘ) with nonnegative integers aᵢ are IDP, with the Minkowski sum ∑aᵢPᵢ also being IDP.
• The Minkowski sum of dilated polytopes ∑nᵢPᵢ is level of index 1 if each nᵢ ≥ dim(Pᵢ)+1, confirming a result from Higashitani (2016).
• The Cayley sum of dilated polytopes n₁P₁*⋯*nₘPₘ is level of index m if each nᵢ ≥ dim(Pᵢ)+1, providing a positive answer to Question 1.5(2).
• The bound nᵢ ≥ dim(Pᵢ)+1 is optimal for the level property, as shown by a counterexample where nᵢ = dim(Pᵢ) fails to yield levelness.
• The IDP property does not hold for Cayley sums under the weaker condition nᵢ ≥ dim(Pᵢ), as demonstrated by a counterexample with three 1-dimensional polytopes whose Cayley sum is not IDP.
• The paper proves that any lattice polytope P with dim(P)=d satisfies that nP is 2-convex-level for all n ≥ d+1, which is essential for establishing the level property of dilated Cayley sums.