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

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