I have several doubts about the proof of Lemma 62 in Petersen's Riemannian Geometry book (pp. 355-356 in the second edition).
Why is $f\le d(\cdot,\partial A)$? One could argue that a segment from $q\in A$ to $H$ should intersect $\partial A$, but a priori this segment could well not belong to $A$.
Secondly, I'd like to see a proof of the second statement (the smoothness at $\sigma(t)$ for $t<a$).
The following part is clear:
Then Petersen studies what happens at $q=\sigma(a)$:
What? By definition $f_t(z)=0$. I think he meant $d(x,\partial A)=d(x,z)+d(z,\partial A)$. But in general $d(x,z)\ge f_t(x)$, so we cannot conclude that $d(x,z)+d(z,\partial A)\le f_t(x)+t$!
The final part is clear:
Any partial answer is appreciated.