On the uniqueness of the critical point of $\psi_\Omega$

Abstract

We prove that for any bounded convex domain $\Omega \subset \mathbb{R}^n$, the function \begin{equation*} \psi_\Omega(\xi) = \int_{\mathbb{R}^n\setminus\Omega} \frac{\mathrm{d}x}{|x-\xi|^{2n}}, \quad \xi\in\Omega, \end{equation*} has exactly one critical point. This confirms an conjecture proposed by Clapp, Pistoia and Salda\~na in [J. Math. Pures Appl. 205 (2026), 103783]. The proof uses a spherical coordinates representation to write $\psi_\Omega$ as an integral of the distance function $\rho(\xi,\omega)$. This approach is not limited to $\psi_\Omega$. Instead, it provides a general framework for analyzing a broad class of functionals involving the boundary distance. We also examine non-convex domains. In particular, a single annulus exhibits a full circle of critical points, while multiple concentric annuli produce finitely many critical spheres. These examples show that the convexity hypothesis is essential for the uniqueness conclusion. The method developed here for handling spherical integrals involving the distance function is likely to be useful in other geometric and analytic contexts.