## Electronic Communications in Probability

### Energy optimization for distributions on the sphere and improvement to the Welch bounds

Yan Shuo Tan

#### Abstract

For any Borel probability measure on $\mathbb{R} ^n$, we may define a family of eccentricity tensors. This new notion, together with a tensorization trick, allows us to prove an energy minimization property for rotationally invariant probability measures. We use this theory to give a new proof of the Welch bounds, and to improve upon them for collections of real vectors. In addition, we are able to give elementary proofs for two theorems characterizing probability measures optimizing one-parameter families of energy integrals on the sphere. We are also able to explain why a phase transition occurs for optimizers of these two families.

#### Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 43, 12 pp.

Dates
Accepted: 7 July 2017
First available in Project Euclid: 15 August 2017

https://projecteuclid.org/euclid.ecp/1502762749

Digital Object Identifier
doi:10.1214/17-ECP73

Mathematical Reviews number (MathSciNet)
MR3693769

Zentralblatt MATH identifier
1378.60049

#### Citation

Tan, Yan Shuo. Energy optimization for distributions on the sphere and improvement to the Welch bounds. Electron. Commun. Probab. 22 (2017), paper no. 43, 12 pp. doi:10.1214/17-ECP73. https://projecteuclid.org/euclid.ecp/1502762749

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