A Tight Lower Bound for Convexly Independent Subsets of the Minkowski Sums of Planar Point Sets

Recently, Eisenbrand, Pach, Rothvoß, and Sopher studied the function M(m, n), which is the largest cardinality of a convexly independent subset of the Minkowski sum of some planar point sets P and Q with |P | = m and |Q| = n. They proved that M(m, n) = O(m2/3n2/3+m+n), and asked whether a superlinear lower bound exists for M(n, n). In this note, we show that their upper bound is the best possible apart from constant factors.