Log-optimal points on the sphere
Complex Analysis Seminar
Indiana University, Bloomington, IN
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.
minimal energy problems, optimal configurations, potentials
Analysis | Mathematics
Peter D. Dragnev (2012).
Log-optimal points on the sphere. Presented at Complex Analysis Seminar, Indiana University, Bloomington, IN.