Abstract. Let [Formula: see text] and let [Formula: see text] and [Formula: see text] be convex polytopes in [Formula: see text]. Denote by [Formula: see text] the set of midpoints of pairs [Formula: see text] with [Formula: see text], a vertex of [Formula: see text], and [Formula: see text], a vertex of [Formula: see text], such that [Formula: see text]. In the case [Formula: see text] and [Formula: see text], sharp lower bounds on [Formula: see text] in terms of the number of vertices of [Formula: see text] were given by Erdős, Furedi, and Fishburn in the early 1990s. Recently, Chestnut, Hildebrand, and Zenklusen gave nontrivial lower bounds on [Formula: see text] in terms of the number of vertices of [Formula: see text] for [Formula: see text]. On the one hand, the main statement of this paper generalizes the Erdős, Furedi, and Fishburn result for distinct convex polytopes. On the other hand, the main result of this paper improves the lower bounds obtained by Chestnut, Hildebrand, and Zenklusen for arbitrary [Formula: see text]. The main statement of this paper is a consequence of an additive combinatorics result. More specifically, it is a consequence of a Freiman cube lemma-type result which holds in arbitrary commutative groups, and it is interesting in its own right. Also, using other techniques, in this paper a slightly better bound of [Formula: see text] is reached when [Formula: see text]. Finally, our results have an application in a quantitative Helly-type theorem.

Abstract. Let [Formula: see text] be a commutative group and [Formula: see text] and [Formula: see text] be nonempty finite subsets of [Formula: see text] and [Formula: see text]. Kneser’s theorem is a fundamental result in additive number theory, and it establishes that [Formula: see text]. For any subset [Formula: see text] of [Formula: see text], write [Formula: see text]. For any [Formula: see text], set [Formula: see text]. An important problem in additive number theory is to find a Kneser-type theorem for the restricted sumsets [Formula: see text]. In particular, more than 20 years ago V. Lev proved that if [Formula: see text] [Formula: see text], for all [Formula: see text] (resp., [Formula: see text]) there is at most one [Formula: see text] (resp., [Formula: see text]) such that [Formula: see text] (resp., [Formula: see text]), and [Formula: see text]; then [Formula: see text]. In the same paper, Lev proposed as a problem to improve [Formula: see text] to something of the form [Formula: see text] with [Formula: see text] whenever [Formula: see text]. Lev’s problem has been solved for some particular groups and some specific subsets [Formula: see text] of [Formula: see text]. However, it remains open for arbitrary groups and arbitrary large subsets [Formula: see text] of [Formula: see text]. Here, as a consequence of the main result of this paper, it is shown that if we take [Formula: see text] instead of [Formula: see text] in the lower bound of [Formula: see text], then indeed we can take as the coefficient of [Formula: see text] something of the form [Formula: see text] with [Formula: see text] whenever [Formula: see text].