Estimating the size of the sum of two sets is a widely investigated topic in various branches of mathematics. In this talk we present results related to a problem of this kind:
For any compact set $A$ in the n-dimensional Euclidean space and any positive integer $k$, let $A_k$ denote its $k$-th average, that is, the set $(A+A+...+A)/k$, where the numerator consists of the Minkowski sum of $A$ with itself $k$ times. It was observed by Shapley, Folkmann and Starr in 1969 that as $k$ tends to infinity, $A_k$ approaches the convex hull $\mathrm{conv}(A)$ of $A$ with respect to Hausdorff distance. Furthermore, $A_1=A$ and $A_k \subseteq \mathrm{conv}(A)$ trivially holds for all values of $k$. The volume deficit of $A_k$ is defined as $V(\mathrm{conv}(A) - A_k)=V(\mathrm{conv}(A))-V(A_k)$. This quantity was examined by Fradelizi, Madiman, Marsiglietti and Zvavitch in 2018, who showed that the sequence $V(A_k)$ is increasing if $A$ is convex, or if $n=1$. Furthermore, they proved that for every value $k>1$, there is a dimension $n(k)$ and an $n(k)$-dimensional starlike set $A$ satisfying $V(A_k) > V(A_{k+1})$. In this talk we show that for any $k \geq n-1$, we have $V(A_k) \leq V(A_{k+1})$ for any starlike set $A$. It is a joint project with M. Fradelizi and A. Zvavitch.