A graph has strong convex dimension 2 if it admits a straight-line drawing in the plane such that its vertices form a convex set and the midpoints of its edges also constitute a convex set. Halman, Onn, and Rothblum conjectured that graphs of strong convex dimension 2 are planar and therefore have at most 3n − 6 edges. We prove that all such graphs have indeed at most 2n − 3 edges, while on the other hand we present an infinite family of non-planar graphs of strong convex dimension 2. We give lower bounds on the maximum number of edges a graph of strong convex dimension 2 can have and discuss several natural variants of this graph class. Furthermore, we apply our methods to obtain new results about large convex sets in Minkowski sums of planar point sets – a topic that has been of interest in recent years.