On the Increase of Capacity with Asymmetry

Abstract

Let E be a compact, starlike set in ℝN, N ≥2, that is very close to a ball B of the same area or volume. This paper presents inequalities, for logarithmic capacity if N = 2 or for capacity if N ≥ 3, of the form \(lcap\ E \geq exp \lbrace K_2\alpha(E)^2\rbrace\ lcap\ \overline B\) or \(cap\ E \geq \lbrace 1+K_N\alpha(E)^2\rbrace\ cap\ \overline B\) where α(E) is a modulus of asymmetry that measures the departure of the shape of E from that of B. The results are far less general than those of Hansen and Nadirashvili for N = 2 and those of Hall, Hayman and Weitsman for N ≥ 3, but (for the particular sets considered) the present inequalities are somewhat sharper.