Let $M$ be a Riemannian manifold with distance function $d$, $C \subset M$ a geodesically convex set, $a=(a_i)_{i=1}^n \in C^n$, $W \in \mathbb{R}_{\geq 0}^{n \times n}$ and $J\colon C^n \rightarrow \mathbb{R}$ be defined by $$J(u):=\sum_{i=1}^n d(a_i,u_i)^2 +\sum_{i,j}^n W_{i,j} d(u_i,u_j)^2 .$$ Does $J$ poses a unique critical point? This problem interests me because of some applications in total variation regularization. There, I want to take $C=S^2_{\geq 0} \subset S^2$, the positive octant of the two dimensional sphere, for denoising of color images, or $M=SPD(3)$ for denoising of diffusion tensor images. For Euclidean spaces, one can use convexity arguments or directly study the linear equations $DJ/du_i=0$ to prove uniqueness of a critical point. For manifolds of nonpositive curvature, such as $SPD(n)$ one can as well use convexity arguments (the squared distance function is convex on manifolds of non-positive curvature). However, for manifolds of positive curvature, such as the sphere, the squared distance function is not convex. WLOG we can assume $W$ symmetric. For the sphere I could prove uniqueness if $$\sum_{j\neq i} W_{i,j} <\frac{2(\pi-2)}{\pi^2+4}$$ for all $i$ by defining a retraction. However I wonder if there is some way to extend the result for arbitrary large weights $W$.

# Does this squared distance functional have a unique critical point on geodesically convex manifolds?

## 1 Answer

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Found a proof for the half sphere by showing that the functional is convex at every critical point and using poincare-hopf theorem.