Large convexly independent subsets of Minkowski sums

Let E-d (n) be the maximum number of pairs that can be selected from a set of n points in R-d such that the midpoints of these pairs are convexly independent. We show that E-2(n) >= Omega (n root 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 left perpendicular1/3n(2)right perpendicular <= E-3(n) <= 3/8n(2) + O(n(3/2)). Let W-d(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 W-2(n) = Theta(E-2(n)) and for d >= 3 that W-d(n) similar to 1/2 (1-1/a(d)) n(2), where a(d) is an element of N is related to strictly antipodal families. In fact we show that the same asymptotics hold with out the requirement that the unit distance pairs form pairwise nonparallel segments,and also if diameter pairs are considered instead of unit distance pairs.