The Kneser–Poulsen Conjecture for Special Contractions

The Kneser–Poulsen conjecture states that if the centers of a family of N unit balls in (Formula presented.) are contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). We consider two types of special contractions. First, 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. We obtain that a uniform contraction of the centers does not decrease the volume of the intersection of the balls, provided that (Formula presented.). Our result extends to intrinsic volumes. We prove a similar result concerning the volume of the union. Second, a strong contraction is a contraction in each coordinate. We show that the conjecture holds for strong contractions. In fact, the result extends to arbitrary unconditional bodies in the place of balls. © 2018 Springer Science+Business Media, LLC, part of Springer Natur

From rr-dual sets to uniform contractions

Let $M^d$ denote the $d$-dimensional Euclidean, hyperbolic, or spherical space. The $r$-dual set of given set in $M^d$ is the intersection of balls of radii $r$ centered at the points of the given set. In this paper we prove that for any set of given volume in $M^d$ the volume of the $r$-dual set becomes maximal if the set is a ball. As an application we prove the following. The Kneser-Poulsen Conjecture states that if the centers of a family of $N$ congruent balls in Euclidean $d$-space is contracted, then the volume of the intersection does not decrease. 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. We prove the Kneser-Poulsen conjecture for uniform contractions (with $N$ sufficiently large) in $M^d$.Comment: 8 page

Entropic exercises around the Kneser-Poulsen conjecture

We develop an information-theoretic approach to study the Kneser--Poulsen conjecture in discrete geometry. This leads us to a broad question regarding whether R\'enyi entropies of independent sums decrease when one of the summands is contracted by a $1$-Lipschitz map. We answer this question affirmatively in various cases.Comment: 23 pages, comments welcome! Final version with minor changes, added Corollary 2.8 (linear contractions decrease intrinsic volumes of convex bodies

On the Volume of Boolean expressions of Large Congruent Balls

We consider the volume of a Boolean expression of some congruent balls about a given system of centers in the d-dimensional Euclidean space. When the radius r of the balls is large, this volume can be approximated by a polynomial of r, which will be computed up to an O(r^{d−3}) error term. We study how the top coefficients of this polynomial depend on the set of the centers. It is known that in the case of the union of the balls, the top coefficients are some constant multiples of the intrinsic volumes of the convex hull of the centers. Thus, the coefficients in the general case lead to generalizations of the intrinsic volumes, in particular, to a generalization of the mean width of a set. Some known results on the mean width, along with the theorem on its monotonicity under contractions are extended to the "Boolean analogues" of the mean width

On a Blaschke-Santal\'o-type inequality for rr-ball bodies

Let ${\mathbb E}^d$ denote the $d$-dimensional Euclidean space. The $r$-ball body generated by a given set in ${\mathbb E}^d$ is the intersection of balls of radius $r$ centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke-Santal\'o-type inequality for $r$-ball bodies: for all $0<k< d$ and for any set of given $d$-dimensional volume in ${\mathbb E}^d$ the $k$-th intrinsic volume of the $r$-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.Comment: 5 pages. arXiv admin note: text overlap with arXiv:1810.1188