  A graph has strong convex dimension $2$, if it admits a straight-line drawing
in the plane such that its vertices are in convex position and the midpoints of
its edges are also in convex position. 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 at most $2n-3$ edges
while on the other hand we present a class of non-planar graphs of strong
convex dimension $2$. We also give lower bounds on the maximum number of edges
a graph of strong convex dimension $2$ can have and discuss variants of this
graph class. We apply our results to questions about large convexly independent
sets in Minkowski sums of planar point sets, that have been of interest in
recent years.
