The polarization problem asks to place light sources such that the darkest point has maximal illumination. We consider the sum of (identically scaled) Gaussian functions centered at points of a periodic configurations with n points per period. We prove that, whenever the number of points per period is sufficiently large (depending only on the scale of the Gaussian), the minimal value of this series is maximized if and only if the points are equispaced. The polarization problem is kind of dual to the problem of energy minimization and our result complements the result of Cohn and Kumar on universal optimality of the (scaled) integers. To the best of our knowledge this is the first polarization result for periodic configurations in any Euclidean space R^d.

The proof involves several steps and different techniques: we use the machinery of theta functions, apply the McGehee-Pigno-Smith inequality and take advantage of properties of the Discrete Fourier Transform.

This is joint work with Stefan Steinerberger from the University of Washington, Seattle.

https://univienna.zoom.us/j/64895816787?pwd=L0tHVnBPUkJFQVVSR3Y2QnhVRXRGZz09