Lattice polytopes with distinct pair-sums (Q1596353)

For a lattice polytope \(P\) in \(\mathbb{R}^n\) let \(P\cap \mathbb{Z}^n= \{v_1,\dots, v_N\}\). If the \(N+{N \choose 2}\) points \(2v_1,\dots,2v_N\), \(v_1+ v_2, \dots, v_{N-1} +v_N\) are distinct, then \(P\) is called a ``distinct pairwise sum'' polytope. The authors prove that for such a polytope \(N\leq 2^n\), and, for every \(n\), they construct ``distinct pairwise sum'' polytopes in \(\mathbb{R}^n\) which contain \(2^n\) lattice points. The notion of a ``distinct pairwise sum'' polytope was motivated by the study of the representation of polynomials as a sum of squares of polynomials.