In this presentation we prove that the equilibrium measure of a finite union of intervals on the real line or arcs on the unit circle has convex density. This is true for both, the classical logarithmic case, and the Riesz case. The electrostatic interpretation is the following: if we have a finite union of subintervals on the real line, or arcs on the unit circle, the electrostatic distribution of many “electrons” will have convex density on every subinterval. Applications to external field problems and constrained energy problems are discussed.
This is Joint work with David Benko - University of South Alabama and Vilmos Totik - University of Szeged and University of South Florida.