By means of Schwarz symmetrization and Pólya-Szegő inequality one can easily prove that the smallest capacity for the sets of the given volume is achieved solely by the balls.
One may wonder if the balls are stable minimizers. Is it true that if the set has the capacity close to the one of the ball then the set itself is close to the ball in some appropriate sense?
The answer is yes. The first result in that direction is contained in the work of Hall, Hayman and Weitsman (1991). In the same paper they conjecture that a certain improved version of their inequality holds.
We provide a positive answer to this conjecture. In this talk we will discuss the proof, which is based on Selection Principle, the technique introduced by Cicalese and Leonardi in 2012.
This is a joint work with Guido De Philippis and Michele Marini.