On uniform contractions of balls in Minkowski spaces

2019. 05. 21. 10:30
Bezdek Károly

Let $N$ balls of the same radius be given in a $d$-dimensional real normed vector space, i.e., in a Minkowski $d$-space. Then apply a uniform contraction to the centers of the $N$ balls without changing the common radius. Here a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. The main result of this talk states that a uniform contraction of the centers does not increase (resp., decrease) the volume of the union (resp., intersection) of $N$ balls in Minkowski $d$-space, provided that $N\geq 2^d$ (resp., $N\geq 3^d$ and the unit ball of the Minkowski $d$-space is a generating set in $\mathbb{R}^d$).