I am using concepts from this paper. Let $M$ be a Riemannian manifold, $D \subset M$ be a compact geodesically convex subset, and $f:D \to \mathbb{R}$ be a differentiable function. We say $f$ has $L$-Lipschitz gradient if for any $x,y \in D$, $$f(y) \leq f(x) + \langle \nabla f(x) ,\log_x(y) \rangle_x + \frac{L}{2} d(x,y).$$ Here, geodesic convex means for any $x,y \in D$, a unique minimizing geodesic segment $\gamma:[0,1] \to M$ starting at $x$ and ending at $y$ is entirely contained in $D$.
Once again, let $D \subset M$ be compact and geodesic convex. Fix $y \in M$. I am interested in the function $f(x) := d(x,y)^2$ for $x \in D$. My question is, does $f$ have $L$-Lipschitz gradient?