Description
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 conv(A) of A with respect to Hausdorff distance. Furthermore, $A_1=A$ and$ A_k \subseteq conv(A)$ trivially holds for all values of k. The volume deficit of $A_k$ is defined as $V(conv(A) - A_k)=V(conv(A))-V(A_k)$. 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.