On the Kneser–Poulsen conjecture in elliptic space

Abstract

The Kneser–Poulsen conjecture claims that if some balls of Euclidean space are rearranged in such a way that the distances between their centers do not increase, then neither does the volume of the union of the balls. A special case of the conjecture, when the balls move continuously in such a way that the distances between the centers (weakly) decrease during the motion, is known to hold not only in Euclidean, but also in spherical and hyperbolic spaces. In the present paper, we show that this theorem cannot be extended to elliptic space by constructing three smoothly moving congruent balls with centers getting closer to one another in such a way that the volume of the union of the balls strictly increase during the motion. In spite of this counterexample, it is true that n + 1 balls in n-dimensional elliptic space cover maximal volume if the distances between the centers are all equal to the diameter π/2 of the space. The second part of the paper is devoted to the proof of this fact.