Isoperimetric inequalities in potential theory

Given a non-empty bounded domainG in ⁿ,n2, letr₀(G) denote the radius of the ballG₀ having center 0 and the same volume asG. The exterior deficiencyd_e(G) is defined byd_e(G)=r_e(G)/r₀(G)–1 wherer_e(G) denotes the circumradius ofG. Similarlyd_i(G)=1–r_i(G)/r₀(G) wherer_i(G) is the inradius ofG. Various isoperimetric inequalities for the capacity and the first eigenvalue ofG are shown. The main results are of the form CapG(1+cf(d_e(G)))CapG₀ and ₁(G)(1+cf(d_i(G)))₁(G₀),f(t)=t³ ifn=2,f(t)=t³/(ln 1/t) ifn=3,f(t)=t^(n+3)/2 ifn4 (for convex G and small deficiencies ifn3).