Few products, many h-fold sums

Improving upon a technique of Croot and Hart, we show that for every $h$, there exists an $\epsilon > 0$ such that if $A \subseteq \mathbb{R}$ is sufficiently large and $|A.A| \le |A|^{1+\epsilon}$, then $|hA| \ge |A|^{\Omega(e^{\sqrt{c\log{h}}})}$.Comment: 19 page

Topological obstructions for vertex numbers of Minkowski sums

We show that for polytopes P_1, P_2, ..., P_r \subset \R^d, each having n_i \ge d+1 vertices, the Minkowski sum P_1 + P_2 + ... + P_r cannot achieve the maximum of \prod_i n_i vertices if r \ge d. This complements a recent result of Fukuda & Weibel (2006), who show that this is possible for up to d-1 summands. The result is obtained by combining methods from discrete geometry (Gale transforms) and topological combinatorics (van Kampen--type obstructions) as developed in R\"{o}rig, Sanyal, and Ziegler (2007).Comment: 13 pages, 2 figures; Improved exposition and less typos. Construction/example and remarks adde