ISOPERIMETRIC-INEQUALITIES IN POTENTIAL-THEORY

Given a non-empty bounded domain G in R(n), n greater-than-or-equal-to 2, let r0(G) denote the radius of the ball G0 having center 0 and the same volume as G. The exterior deficiency d(e)(G) is defined by d(e)( G) = r(e)(G)/r0(G) - 1 where r(e)(G) denotes the circumradius of G. Si milarly d(i)(G) = 1 - r(i)(G)/r0(G) where r(i)(G) is the inradius of G . Various isoperimetric inequalities for the capacity and the first ei genvalue of G are shown. The main results are of the form Cap G greate r-than-or-equal-to (1 + cf(d(e)(G))) greater-than-or-equal-to Cap G0 a nd lambda1(G) greater-than-or-equal-to (1 + cf(d(i)(G)))lambda1(G0), f (t) = t3 if n = 2, f(t) = t3/(ln 1/t) if n = 3, f(t) = t(n+3)/2 if n g reater-than-or-equal-to 4 (for convex G and small deficiencies if n gr eater-than-or-equal-to 3).