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We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions.
Moments of isotropic measures and optimal projective codes
A. Glazyrin · 1904
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Energy on spheres and discreteness of minimizing measures
D. Bilyk, A. Glazyrin, R. Matzke, J. Park, and O. Vlasiuk · 1908
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M. Mohammadpour and S. Waldron
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Hilbert distances and positive definite functions
S. Bochner · 1941
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Positive definite functions on spheres
I. J. Schoenberg · 1941
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Two-Point Homogeneous Spaces
H.-C. Wang · 1952
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Zur ebenen Oktavengeometrie
H. Freudenthal · 1953
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Distributions of positive mass, which maximize a certain generalized energy integral
G. Björck · 1956
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Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters
R. Gangolli · 1967
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Some seventh degree integration formulas for symmetric regions
A. Stroud · 1967
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Special Functions and the Theory of Group Representations
N.J. Vilenkin · 1968
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An algebraic approach to the association schemes of coding theory
P. Delsarte · 1973
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Equiangular Lines
P.W.H. Lemmens and J.J. Seidel · 1973
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A survey of two-graphs
J. J. Seidel · 1973
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New estimates for the closest packing of spheres in n n -dimensional Euclidean space
V. M. Sidel’nikov · 1974
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Orthogonal Polynomials
G. Szegő · 1975
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Spherical codes and designs
P. Delsarte, J. M. Goethals, and J. J. Seidel · 1977
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Tight spherical designs I
E. Bannai and R. Damerell · 1979
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Tight spherical designs II
E. Bannai and R. Damerell · 1980
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t-designs in projective spaces
S. G. Hoggar · 1982
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Projection Bodies
J. Bourgain and J. Lindenstrauss · 1988
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Tight t t -designs and squarefree integers
E. Bannai and S. Hoggar · 1989
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Tight 4 4 - and 5 5 -designs in projective spaces
S. G. Hoggar · 1989
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A construction of spherical 2 2 -designs
Y. Mimura · 1990
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Designs as maximum codes in polynomial metric spaces
V. I. Levenshtein · 1992
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Sums of Even Powers of Real Linear Forms
B. Reznick · 1992
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Minimum potential energy of a point system of charges
V. A. Yudin · 1992
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Constructive Approximation
R. A. DeVore and G. G. Lorentz · 1993
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Convex Bodies: the Brunn-Minkowsi theory
R. Schneider · 1993
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On the Korkin-Zolotarev construction
A. V. Kolushov and V. A. Yudin · 1994
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Irreducible finite integral matrix groups of degree 8 and 10
B. Souvignier · 1994
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Cubature for the Sphere and the Discrete Spherical Harmonic Transform
M. Taylor · 1995
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Extremal dispositions of points on the sphere
Fourier analysis in convex geometry
A. Koldobsky · 2005
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Lower bounds on the maximum cross correlation of signals
L. Welch · 2006
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The D 4 D_{4} root system is not universally optimal
H. Cohn, J. Conway, N. Elkies, and A. Kumar · 2007
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Universally optimal distribution of points on spheres
H. Cohn and A. Kumar · 2007
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Harmonic Analysis on Commutative Spaces
J. Wolf · 2007
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Unitary Reflection Groups
G. Lehrer and D. Taylor · 2009
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On tight projective designs
Y. I. Lyubich · 2009
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A. V. Kolushov and V. A. Yudin · 1997
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Blaschke-Santaló inequalities
E. Lutwak and G. Zhang · 1997
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Lower bounds for spherical designs
V. A. Yudin · 1997
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Universal Bounds for Codes and Designs
V. I. Levenshtein · 1998
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The shadow theory of modular and unimodular lattices
E. Rains and N. J. A. Sloane · 1998
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A spherical code
N. N. Andreev · 1999
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Symmetric informationally complete positive- operator-valued measures: a new computer study
A. J. Scott, M. Grassl · 2010
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Stability of ring patterns arising from two-dimensional particle interactions
T. Kolokolnikov, H. Sun, D. Uminsky, and A. L. Bertozzi · 2011
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Grundzüge einer nichtkommutativen Designtheorie
G. Zauner · 2011
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Minimization of the probabilistic p-frame potential
M. Ehler and K. A. Okoudjou · 2012
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Predicting pattern formation in particle interactions
J. H. Von Brecht, D. Uminsky, T. Kolokolnikov, and A. L. Bertozzi · 2012
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Nonlocal interactions by repulsive–attractive potentials: radial ins/stability
D. Balagué, J. Carrillo, T. Laurent, and G. Raoul · 2013
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On the Support of Minimizers of Causal Variational Principles
F. Finster and D. Schiefeneder · 2013
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Systems of Imprimitivity for the Clifford Group
D.M. Appleby, I. Bengtsson, S. Brierley, A. Ericsson, M. Grassl, and J.A. Larsson · 2014
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Remarks on hilbert identities, isometric embeddings, and invariant cubature
H. Nozaki and M. Sawa · 2014
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Nonlocal interaction equations in environments with heterogeneities and boundaries
L. Wu and D. Slepčev · 2015
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Optimal simplices and codes in projective spaces
H. Cohn, A. Kumar, and G. Minton · 2016
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Geometry of minimizers for the interaction energy with mildly repulsive potentials
J. Carrillo, A. Figalli, and F. S. Patacchini · 2017
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Computational Mathematics with SageMath
P. Zimmermann, A. Casamayou, N. Cohen, G. Connan, T. Dumont, L. Fousse, F. Maltey, M. Meulien, M. Mezzarobba, C. Pernet, N. M. Thiéry, E. Bray, J. Cremona, M. Forets, A. Ghitza, and H. Thomas · 2018
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Singular Support of Minimizers of the Causal Variational Principle on the Sphere
L. Bäuml, F. Finster, D. Schiefeneder, and H. von der Mosel · 2019
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D. Bilyk and F. Dai · 2019
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Universal optimal configurations for the p-frame potentials
X. Chen, V. Gonzales, E. Goodman, S. Kang, and K. A. Okoudjou · 2020
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Experimental survey of discrete minimizers of the p p -frame energy
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Problems with a lot of potential: energy optimization on compact spaces
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