Lattice polytopes with distinct pair-sums
  Let P be a lattice polytope in R^n, and let P \cap Z^n = {v_1,...,v_N}. If the N + \binom N2 points 2v_1,...,2v_N; v_1+v_2,...v_{N-1}+v_N are distinct, we say that P is a "distinct pair-sum" or "dps" polytope. We show that, if P is a dsp polytope in R^n, then N \le 2^n, and, for every n, we construct dps polytopes in R^n which contain 2^n lattice points. We also discuss the relation between dps polytopes and the study of sums of squares of real polynomials. 
M. D. Choi (University of Toronto), T. Y. Lam (University of   California, Berkeley), Bruce Reznick (University of Illinois ...