Large Convexly Independent Subsets of Minkowski Sums

Let Ed(n) be the maximum number of pairs that can be selected from a set of n points in Rd such that the midpoints of these pairs are convexly independent. We show that E2(n) > Ω(n √ log n), which answers a question of Eisenbrand, Pach, Rothvoß, and Sopher (2008) on large convexly independent subsets in Minkowski sums of finite planar sets, as well as a question of Halman, Onn, and Rothblum (2007). We also show that ⌊13n⌋ 6 E3(n) 6 38n + O(n3/2). Let Wd(n) be the maximum number of pairwise nonparallel unit distance pairs in a set of n points in some d-dimensional strictly convex normed space. We show that W2(n) = Θ(E2(n)) and for d > 3 that Wd(n) ∼ 12 (