Proving a metric ball is strictly geodesically convex on a Hadamard manifold 
Let $M$ be a Hadamard manifold (complete, connected and with sectional curvature $K \leq 0$) and fix $p \in M, r \geq 0$. Then every metric ball $B_r(p)$ of $M$ is strictly geodesically convex, that is, any geodesic segment connecting two points of $B_r(p)$ is entirely contained in $B_r(p)$.

Intuitively I know that for a geodesic segment $\gamma: [0, 1] \to M$,  I should show that $d(\gamma(t), p)$ achieves its maximum at $\gamma(1)$ (that would finish the problem). However, I haven't been able to recognize what it is about Hadamard manifolds that would force this to be the case. In another post it was suggested this could be a consequence of Rauch's comparison theorem but I can't see how either. I'd appreciate any help. Thanks in advance!
 A: First of all, you are slightly misstating the definition of a strictly closed subset: What you defined is the ordinary convexity. Strict convexity of a closed set $C$ means that, in addition to the convexity $C$, the boundary of $C$ contains no nondegenerate geodesic segments.
Now, to the question at hand. I do not have do Carmo's book in front of me, but Jorgen Jost in
Jost, Jürgen, Riemannian geometry and geometric analysis, Universitext. Berlin: Springer (ISBN 978-3-540-77340-5/pbk). xiii, 583 p. (2008). ZBL1143.53001.
proves, as an application of the Rauch Comparison Theorem, a number of geometric properties of the Riemannian distance function $d$ on Hadamard manifolds $X$, sections 4.6-4.8 of Chapter 4. For instance, he proves that
the function $f(x):= \frac{1}{2}d^2(p,x)$ satisfies the inequality
$$
Hess_f(x)(v,v)\ge ||v||^2
$$
for every $x\in X$ and $v\in T_xX$ (Lemma 4.8.2). Here $Hess$ denotes Hessian. This establishes a strong form of convexity of the function $f$. I will leave it to you to derive strict convexity of closed balls in $X$ from this inequality. You can also easily derive the desired conclusion from Lemma 4.8.3: If $\gamma(t)$ is a geodesic in $X$, then
$$
d^2(p, \gamma(t))\le (1-t) d^2(p, \gamma(0)) + t d^2(p, \gamma(1)) - t(1-t)d^2(\gamma(0), \gamma(1)).
$$
The ultimate form of convexity, however, is in Theorem 4.8.2, that establishes convexity of the function $d^2(c_1(t), c_2(t))$ for any pair of geodesics $c_1, c_2$ in $X$.
By the way, I really like Jost's book for other things such as treatment of connections and curvature on vector bundles, etc.
