On the Volume of Boolean Expressions of Balls – A Review of the Kneser–Poulsen Conjecture

Waist of Balls in Hyperbolic and Spherical Spaces

“…In case the lower and the upper Minkowski content for X and f (X) coincide, we just say that the Minkowski content decreases under a 1-Lipschitz map. The proof of this theorem consists of essentially known results and a particular case of it, for the Euclidean space, is proved independently in [8,Proposition 4.1].…”

Entropic exercises around the Kneser–Poulsen conjecture

“…The formulation with different radii, as considered by some authors (e.g., [5,14]), is not dealt with in our work. We refer the reader to the book [4], or recent surveys [15,31] for a more detailed account.…”

The face structure of ball polyhedra and soft density of ball packings in R3

“…One of the motivations for studying ball polyhedra was the Kneser-Poulsen conjecture which states that if a finite set of centers of balls in R n is rearranged in a way that the distance between each pair of centers does not decrease, then the volume of the intersection (resp., union) does not increase (resp., decrease) (cf. [8,9,13]).…”