# Optimal measures for $p$-frame energies on spheres

@article{Bilyk2022OptimalMF, title={Optimal measures for \$p\$-frame energies on spheres}, author={Dmitriy Bilyk and Alexey A. Glazyrin and Ryan Matzke and Josiah Park and O. V. Vlasiuk}, journal={Revista Matem{\'a}tica Iberoamericana}, year={2022} }

We provide new answers about the placement 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…

## 9 Citations

Energy on spheres and discreteness of minimizing measures

- Mathematics, PhysicsJournal of Functional Analysis
- 2021

In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete…

Constructing high order spherical designs as a union of two of lower order

- Mathematics
- 2019

We show how the variational characterisation of spherical designs can be used to take a union of spherical designs to obtain a spherical design of higher order (degree, precision, exactness) with a…

Repeated Minimizers of p-Frame Energies

- Computer Science, MathematicsSIAM J. Discret. Math.
- 2020

The problem of minimizing the p-frame energy of X is connected to another optimization problem, so giving new lower bounds for such energies, and it is proved that for $1\leq m<d$, a repeated orthonormal basis construction of $N=d+m$ vectors minimizes the energy over an interval.

Geometric Designs and Rotatable Designs I

- Computer Science, MathematicsGraphs Comb.
- 2021

A unified mathematical description of various classes of factorial designs, including Box–Hunter solid designs, Box–Behnken designs, central composite designs and Plackett–Burman designs, in the modern framework of the cubature theory is provided.

Gradient Flows for Frame Potentials on the Wasserstein Space

- Mathematics
- 2018

In this paper we bring together some of the key ideas and methods of two very lively fields of mathematical research, frame theory and optimal transport, using the methods of the second to answer…

Bounds on Antipodal Spherical Designs with Few Angles

- MathematicsThe Electronic Journal of Combinatorics
- 2021

A finite subset $X$ on the unit sphere $\mathbb{S}^d$ is called an $s$-distance set with strength $t$ if its angle set $A(X):=\{\langle \mathbf{x},\mathbf{y}\rangle : \mathbf{x},\mathbf{y}\in X,…

Potential theory with multivariate kernels

- Mathematics
- 2021

In the present paper we develop the theory of minimization for energies with multivariate kernels, i.e. energies, in which pairwise interactions are replaced by interactions between triples or, more…

Discreteness of the minimizers of weakly repulsive interaction energies on Riemannian manifolds

- Mathematics
- 2020

It is shown that the supports of measures minimizing weakly repulsive energies on Riemannian manifolds with sectional curvature bounded below do not have concentration points. This extends the…

On the Search for Tight Frames of Low Coherence

- Computer Science, MathematicsArXiv
- 2020

This work introduces a projective Riesz-kernel for the unit sphere and investigates properties of $N$-point energy minimizing configurations for such a kernel, showing that these configurations form frames that are well-separated and have low coherence.

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