Problem: Let $S:M\rightarrow N$ be a submetry between connected Riemannian manifolds. So $S(B(p,r))=B(S(p),r)$ for all $p$ and $r>0$. Then, we know that for every $n\in N$, there exists some $r(n)>0$ such that for any $s\leq r$, $\overline{B}_{s}(n)$ is strictly convex in $N$. Now, I was wondering if, say, $N$ is compact (or $M$ or both) then can we find an $R>0$ such that for every $n$ and every $R'<R$, $\overline{B}(n,R')$ is strictly convex in $N$.
That is, can we take the radius independent of the points when compactness involved? It seems to me yeah but not sure how. I'd like a purely metric proof if possible