Let $(\mathcal{M},g)$ be a smooth Riemannian manifold. We say that an (open) subset $U \subset \mathcal{M}$ is strongly convex if and only if every pair $p$, $q \in U$ has a unique geodesic of minimum length connecting them, and this geodesic lies completely in $U$. Note that besides the unique minimizing geodesic, there may be other geodesics between $p$ and $q$ which do not minimize length.
I have seen multiple posts which claim that any strongly convex set is totally normal, i.e. for every $p \in U$ we have that $\exp_p: \exp_p^{-1}(U) \subset T_p\mathcal{M} \to U$ is a diffeomorphism, see for example here and here. However, I can't find a reference where this is shown and failed to prove it myself.
I am aware that it suffices to show the injectivity of $\exp_p$ on $\exp_p^{-1}(U)$ for every $p \in U$. In fact, assuming we already know that $\exp_p$ is locally injective$^1$, we can deduce that any geodesic $\gamma : \lbrack 0, b \rbrack \to \mathcal{M}$ lying in $U$ cannot seize to be minimizing: Otherwise, there would exist a cut time $0 < c < b$ such that $\gamma$ is no longer minimizing past $\lbrack 0, c\rbrack$ and one can construct at least two distinct minimizing geodesics connecting $p$ with $\gamma(c)$. $^2$
Therefore, all geodesics lying in $U$ must be minimizing and an argument similar to this answer then yields the claim.
In summary: I am looking for an argument as to why $\exp_p$ is (locally) injective on the preimage $\exp_p^{-1}(U)$ if $U$ is a strongly convex set containing $p$.
$$ $$ $\small{^1 : \text{I.e. for every} \ v \in \exp_p^{-1}(U) \ \text{there exists an open neighborhood} \ V \subset \exp_p^{-1}(U) \ \text{of} \ v \ \text{such that} \ \ \\exp_p\vert_V \ \text{is injective}.}$ $\small{^2 : \text{C.f. Proposition} \ 10.32 \ \text{b)} \ \text{in Jack Lee's }}$ Introduction to Riemannian Manifolds
