On Sumsets and Convex Hull

One classical result of Freiman gives the optimal lower bound for the cardinality of $$A+A$$A+A if $$A$$A is a $$d$$d-dimensional finite set in $$\mathbb R^d$$Rd. Matolcsi and Ruzsa have recently generalized this lower bound to $$|A+kB|$$|A+kB| if $$B$$B is $$d$$d-dimensional, and $$A$$A is contained in the convex hull of $$B$$B. We characterize the equality case of the Matolcsi–Ruzsa bound. The argument is based partially on understanding triangulations of polytopes.