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 (20...