On Sumsets and Convex Hull

Discrete & Computational Geometry, Sep 2014

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 $$\mathbb R^d$$. 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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Károly J. Böröczky, Francisco Santos, Oriol Serra. On Sumsets and Convex Hull, Discrete & Computational Geometry, 2014, 705-729, DOI: 10.1007/s00454-014-9633-2