# Is there a unique minimizing geodesic to a geodesically convex set?

Let $$(\mathcal M,g)$$ be a geodesically complete Riemannian manifold, and $$\mathcal X\subset \mathcal M$$ be a compact, geodesically convex subset.

It is trivial that for any point $$x\in\mathcal M$$, there exists a unique $$y\in\mathcal X$$ minimizing $$d(x,y)$$.

But is there a unique geodesic of minimal length from $$x$$ to $$y$$ in $$\mathcal M$$?

• Could you provide definitions of geodesically complete Riemannian manifold and geodesically convex subset? Assuming the surface of a sphere belong to such sets, if $x$ and $y$ are two antipodal points, then the minimal geodesics are not unique. Commented Oct 1, 2018 at 13:45
• thanks! that's a valid counter-example Commented Oct 1, 2018 at 13:54
• It's not necessarily true that the closest point to $x$ in $\mathcal X$ is unique. For example, let $\mathcal M$ be the unit sphere in $\mathbb R^3$ with its standard metric, let $\mathcal X$ be a small closed arc on the equator (of length less than $\pi$), and let $x$ be the north pole. Then $\mathcal X$ is compact and geodesically convex, but every point of $\mathcal X$ is the same distance from $x$. Commented Oct 1, 2018 at 21:18

In $$\mathbb{S}^n$$, take a small closed ball (i.e. south of the equator) centered at the south pole. There is no unique $$y$$ that minimizes the distance from the north pole (the whole boundary is a constant latitude circle and hence every point there minimizes $$d(N,-)$$).