On the Minimax Spherical Designs

Distributing points on a (possibly high-dimensional) sphere with minimal energy is a long-standing problem in and outside the field of mathematics. This paper considers a novel energy function that arises naturally from statistics and combinatorial optimization, and studies its theoretical properties. Our result solves both the exact optimal spherical point configurations in certain cases and the minimal energy asymptotics under general assumptions. Connections between our results and the L1-Principle Component Analysis and Quasi-Monte Carlo methods are also discussed.