Suppose $(M, g)$ is a Riemannian manifold. For each $x \in M$ define the $\textbf{convexity radius of $M$ at $x$}$, denoted by $\text{conv}(x)$ to be the supremum of all $\epsilon > 0$ s.t. there is a geodesically convex geodesic ball $\mathcal B_{\epsilon}(x)$. By geodesically convex we mean each pair of points in $\mathcal B_\epsilon(x)$ can be connected by a unique minimizing geodesic staying within the ball. How to prove that $\text{conv}(x)$ is continuous?
First, I wanted to prove upper-semicontinuity:
Suppose $\text{conv}(x_n) \ge c$ for a sequence $x_n \to x$. Pick $y, z \in B_c(x)$ and let $d = \max\{\text{d}(y, x)), \text{d}(z, x)\}$. Find $n$ large enough so that $\text{d}(x_n, x) < \frac{1}{2}(c' - d)$ for some $c' \in (d, c)$. Then \begin{align*} \text{d}(y, x_n) &\le \text{d}(y, x) + \text{d}(x, x_n) < d + \frac{1}{2}(c' - d) = \frac{1}{2}(c' + d) < c', \end{align*} and similarly for $z$. It follows that $y, z \in \mathcal B_{c'}(x_n)$ and by hypothesis we can find a unique geodesic $\gamma \in \mathcal B_{c'}(x_n)$ connecting $y$ and $z$ which satisfies \begin{align*} \text{d}(\gamma(t)), x) &\le \text{d}(\gamma(t), x_n) + \text{d}(x_n, x) < c' + \frac{1}{2}(c' - d). \end{align*} If we chose $c'$ close enough to $d$ so that $\frac{1}{2}(c' - d) < c - c'$, we would that $\gamma$ never leaves $B_c(x)$, so all points in this ball can be connected by a minimizing geodesic. If $\alpha : [0, T]$ were another such geodesic, then we could just let $d = \max_{t \in [0, T]} \text{d}(\gamma(t), x)$ and repeat the argument above to show that $\alpha$ is contained in some geodesically convex $\mathcal B_{c'}(x_n)$, and this would in turn force $\alpha = \gamma$ and establish the uniqueness necessary for $B_c(x)$ to be a geodesically convex ball.
Here I run into problems because part of the definition is that $B_c(x)$ must be a geodesic ball, so $\exp_x$ must be a diffeomorphism onto it. I know $\exp_x$ is bijective, but this does not necessarily mean its inverse is smooth. Is there a way to apply the global rank theorem to complete this part?
Also, for lower semicontinuity, all I managed to show is that every ball centered sufficiently close to $x$ can have two of its points connected by a unique minimizing geodesic (not necessarily within the ball itself). How does one prove the lower semicontinuity of $\text{conv}(x)$?
