Integer Decomposition Property of Free Sums of Convex Polytopes

Let P ⊂ R and Q ⊂ R be integral convex polytopes of dimension d and e which contain the origin of R and R, respectively. In the present paper, under some assumptions, the necessary and sufficient condition for the free sum of P and Q to possess the integer decomposition property will be presented. Introduction A convex polytope is called integral if any of its vertices has integer coordinates. Let P ⊂ R and Q ⊂ R be convex polytopes and suppose that 0d ∈ P and 0e ∈ Q, where 0d ∈ R d denotes the origin of R and 0e ∈ R e denotes that of R. We introduce the canonical injections μ : R → R by setting μ(α) = (α, 0e) ∈ R d+e with α ∈ R and ν : R → R by setting ν(β) = (0d, β) ∈ R d+e with β ∈ R. In particular, μ(0d) = ν(0e) = 0d+e, where 0d+e denotes the origin of R . Then μ(P) and ν(Q) are convex polytopes of R with μ(P) ∩ ν(Q) = 0d+e ∈ R . The free sum of P and Q is the convex hull of the set μ(P) ∪ ν(Q) in R. It is written as P ⊕Q. One has dim(P ⊕Q) = dimP + dimQ. For a convex polytope P ⊂ R and for each integer n ≥ 1, we write nP for the convex polytope {nα : α ∈ P} ⊂ R. We say that an integral convex polytope P ⊂ R possesses the integer decomposition property if, for each n ≥ 1 and for each γ ∈ nP ∩ Z, there exist γ, . . . , γ belonging to P ∩ Z such that γ = γ + . . .+ γ. Let P ⊂ R and Q ⊂ R be convex polytopes containing the origin (of R or R). It is then easy to see that if the free sum of P and Q possesses the integer decomposition property, then each of P and Q possesses the integer decomposition property. On the other hand, the converse is not true in general. (See Example 0.3.) The purpose of the present paper is to show the following Theorem 0.1. Let P ⊂ R and Q ⊂ R be integral convex polytopes of dimension d and dimension e containing 0d and 0e, respectively. Suppose that P and Q satisfy Z(P ∩ Z) = Z, Z(Q ∩ Z) = Z and (P ⊕Q) ∩ Z = μ(P ∩ Z) ∪ ν(Q ∩ Z). (1) 2010 Mathematics Subject Classification: 52B20.