#### Title

Log-optimal points on the sphere

#### Document Type

Presentation

#### Presentation Date

Fall 10-25-2012

#### Conference Name

Complex Analysis Seminar

#### Conference Location

Indiana University, Bloomington, IN

#### Peer Review

Invited

#### Abstract

The product of all N(N-1)/2 possible distances for a collection of N points on the circle is maximized when the points are (up to rotation) the N-th roots of unity. There is an elegant elementary proof of this fact. In higher dimensions the problem becomes much more complicated. For example, if the points are restricted to the unit sphere in 3-space, the result is known for N=1-6, and 12. Using the tools of multivariate calculus and linear algebra we will derive a characterization theorem for the stationary points in d-space. We illustrate this theorem with a couple of examples of new optimal configurations in the literature.

#### Keywords

minimal energy problems, optimal configurations, potentials

#### Disciplines

Analysis | Mathematics

#### Opus Citation

Peter D. Dragnev (2012).
*Log-optimal points on the sphere*. Presented at Complex Analysis Seminar, Indiana University, Bloomington, IN.

https://opus.ipfw.edu/math_facpres/92