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 pframe 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.
|Original language||English (US)|
|Number of pages||32|
|Journal||Revista Matematica Iberoamericana|
|State||Published - 2022|
Bibliographical noteFunding Information:
Funding. We express our gratitude to the following organizations, that hosted subsets of the authors during the work on this paper: AIM, ICERM, INI, CIEM, and Georgia Tech. The authors were supported by the following grants: DMS-1665007, DMS-2054606 (DB), DMS-2054536 (AG), the Graduate Fellowship 00039202 (RM), and in part by the grant DMS-1600693 and Tripods grant CCF-1934904 (JP), all from the US National Science Foundation, as well as by the Simons collaboration grant for mathematicians (DB) and by the AMS-Simons travel grant (OV). This work was also supported by EPSRC grant no. EP/K032208/1 (the authors’ visit to INI) and NSF DMS-1439786 (research in groups at ICERM).
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