Sharp inequalities for the Newtonian and logarithmic capacities

Sharp inequalities between the Riesz and logarithmic capacities of compact sets in ${\mathbb{R}}^n$ are established, and the equality conditions are fully characterized. These results naturally extend Pólya and Szegő’s conjecture on the ratio of Newtonian to logarithmic capacities in ${\mathbb{R}}^2$, which was posed in the 1940s.