Sums of finite subsets in \(\mathbb{R}^d\) (Q5920215)

The author proves that if \(B_1,B_2\subset \mathbb{R}^d\) are finite sets such that \(\dim B_2=d\), \(n\) is a positive integer and \(B_1\subseteq n\cdot \mathrm{conv}B_2\), where \(\mathrm{conv}\) stands for the convex hull, then for the sumset \(B_1+B_2\) we have the estimate \[ |B_1+B_2|\geq \frac{n+d}{n}|B_1|-\frac 1n \binom{n+d}{d-1}. \] An application of the induction extension of this result to the sumset \(B_1+B_2+\ldots+B_m\) provides (among other consequences) the positive answer to problem 1.8 posed by \textit{M. Matolcsi} and \textit{I. Z. Ruzsa} [in: Additive number theory. Festschrift in honor of the sixtieth birthday of Melvyn B. Nathanson. New York, NY: Springer, 221--227 (2010; Zbl 1300.11016)].