Optimal measures for p-frame energies on spheres

• DOI: 10.48550/arxiv.1908.00885
• Publication Date: 2022-01-03
• Authors: Bilyk, Dmitriy; Glazyrin, Alexey; Matzke, Ryan; Park, Josiah; Vlasiuk, Oleksandr
• Keywords: Mathematics - Metric Geometry; Mathematics - Classical Analysis and ODEs; Classical Analysis and ODEs (math.CA); FOS: Mathematics; Mathematics - Combinatorics; Metric Geometry (math.MG); Combinatorics (math.CO); 52A40, 52C17 (Primary) 41A05, 42C15, 05B30 (Secondary); 0101 mathematics; 01 natural sciences

We provide new answers about the distribution of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the p -frame energies, i.e., energies with the kernel given by the absolute value of the inner product raised to a positive power p . Application of linear programming methods in the setting of projective spaces allows for describing the minimizing measures in full in several cases: we show optimality of tight designs and of the 600-cell for several ranges of p in different dimensions. Our methods apply to a much broader class of potential functions, namely, those which are absolutely monotonic up to a particular order.