Equal-radius Kneser–Poulsen inequality in dimensions three and higher

Prove the equal-radius Kneser–Poulsen inequality in every dimension d ≥ 3: whenever a labeled configuration of Euclidean ball centers is expanded so that no pairwise distance decreases, show that the volume of the union of equal-radius balls does not decrease.

Background

The paper represents Geometric Kernel Diversity as an integral of equal-radius Euclidean ball-union volumes. Consequently, the Kneser–Poulsen inequality would imply distance monotonicity for GKD for arbitrary admissible radial kernels.

The inequality is established in dimension one and follows from the planar Kneser–Poulsen theorem in dimension two, but the general statement remains unresolved in dimensions at least three. The paper’s unconditional monotonicity theorem for its broader kernel class is obtained by a separate analytic argument, while the appendix records the relationship to this geometric conjecture.