On the convexification effect of vector addition

Időpont: 
2019. 03. 26. 10:30
Hely: 
H306
Előadó: 
Lángi Zsolt

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 Ak 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, Ak approaches the convex hull conv(A) of A with respect to Hausdorff distance. Furthermore, A1=A and Akconv(A) trivially holds for all values of k. The volume deficit of Ak is defined as V(conv(A)Ak)=V(conv(A))V(Ak). This quantity was examined by Fradelizi, Madiman, Marsiglietti and Zvavitch in 2018, who showed that the sequence V(Ak) 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(Ak)>V(Ak+1). In this talk we show that for any kn1, we have V(Ak)V(Ak+1) for any starlike set A. It is a joint project with M. Fradelizi and A. Zvavitch.