Abstract: Let P be a lattice polytope in R n , and let P \cap Z n = {v 1 ,\ldots,v N } . If the N + N\choose 2 points 2v 1 ,\ldots, 2v N ;v 1 +v 2 ,\ldots, 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 dps polytope in R n , then N≤ 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.