Concentric balls for thermal insulation with a radiative boundary law

We consider a thermal insulation energy with a quadratic Dirichlet term and the nonlinear boundary potential associated with radiative heat transfer. Both the heated set and the surrounding domain may vary, with heated volume normalized to that of the unit ball and an upper bound on the total volume. We prove that the minimum is achieved within the family of concentric balls, allowing a smaller outer radius and the bare heated ball. The comparison holds in every dimension \(n\ge2\), for every positive boundary coefficient and every nonnegative ambient parameter. The proof rearranges the boundary potential and retains its singular variation. A concavity property of the radiative law then removes the resulting interior jumps without increasing the energy.