LetB₁ be a ball of radiusr₁ inSⁿ (ℍⁿ), and letB₀ be a smaller ball of radiusr₀ such thatB₀ ⊂B₁. ForSⁿ we considerr₁π. Let u be a solution of the problem- δm = 1 in Ω :=B₁ /B₀ vanishing on the boundary. It is shown that the associated functionalJ (Ω) is minimal if and only if the balls are concentric. It is also shown that the first Dirichlet eigenvalue of the Laplacian on Ω is maximal if and only if the balls are concentric.

Let $\wp 1,\wp 0$ be two regular polygons of 𝑛 sides in a space form $M^2(\kappa)$ of constant curvature $\kappa=0,1$ or $-1$ such that $\wp 0\subset\wp 1$ and having the same center of mass. Suppose $\wp 0$ is circumscribed by a circle 𝐶 contained in $\wp 1$. We fix $\wp 1$ and vary $\wp 0$ by rotating it in 𝐶 about its center of mass. Put $\Omega =(\wp 1\backslash\wp 0)^0$, the interior of $\wp 1\backslash\wp 0$ in $M^2(\kappa)$. It is shown that the first Dirichlet’s eigenvalue $\lambda 1(\Omega)$ attains extremum when the axes of symmetry of $\wp 0$ coincide with those of $\wp 1$.