It is well-known that in a geodesic ball centered at $p$, the radial geodesic between $p$ and $q$ is the unique minimizing curve. I'm trying to follow the proof of this given in Cheeger & Ebin (AMS Chelsea edition, page 8). They seem to take a slightly different approach than usual.
Let $B_r(p)$ be a geodesic ball, and let $c:[0,1]\to M$ be a piecewise smooth curve from $p$ to $\exp_pv\in B_r(p)$. Let $r$ be the radial distance function on the ball. Suppose $r(c(t))\le r$ $\forall t\le t_0$, that is, $c$ is inside the ball for $t\le t_0$. The authors show that $$\tag{1}L(c)\ge r(c(t_0))+\int_{t_0}^1||c'||\,dt,$$ with equality holding iff $c'=\lambda\partial_r$, for $\lambda$ some nonnegative function. They then define $t_1$ as the first $t$ for which $r(c(t_1))=||v||$. This follows from the Intermediate Value Theorem, and I understand this. They then say $$\tag{2}L(c)=||v||+\int_{t_1}^1||c'||\,dt.$$ However, this would imply that $$||v||=\int_0^{t_1}||c'||\,dt=r(c(t_1)),$$ which seems incorrect for general curves.
Is (2) correct? I suspect there should really be a $\ge$ instead of $=$, so it is consistent with (1). In particular, the rest of the proof makes sense with $\ge$, but it is then not clear why (1) needed to be derived.