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

TL;DR

This paper proves the uniqueness of the critical point of a specific boundary integral function for convex domains, confirming a recent conjecture, and introduces a spherical coordinate method applicable to broader geometric functionals.

Contribution

It establishes the uniqueness of the critical point for ta_\u03a9 in convex domains and develops a general spherical coordinate framework for analyzing boundary distance functionals.

Findings

ta_ta has exactly one critical point in convex domains.

Non-convex domains like annuli can have multiple or circle of critical points.

The spherical coordinate approach is broadly applicable to geometric analysis.

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.