Xjyuk

This question stems from the discussion in:

how to define the injectivity radius of manifolds with boundary?

Suppose $(M,g)$ is a compact Riemannian manifold with boundary. In this context, let the injectivity radius of a point $x$ be the minimum distance from $x$ at which there is a point $y$ with more than one length-minimizing geodesic connecting $x$ to $y$.

Is it true that the injectivity radius as defined this way is bounded below by some nonzero value? If so, is there a standard reference for this fact?

In the discussion linked above, the following theorem is referenced:

Corollary 2. If for a complete Riemannian manifold with boundary, M, the
sectional curvatures of the interior and the outward sectional curvatures of the
boundary are no greater than $K$, then $N(p,fracpi2K)$ is open in M and the
distance function from p is convex on $N(p,fracpi2K)$.

Where $N(p,fracpi2K)$ is the set of points connected to $p$ by a unique geodesic of lenght $fracpi2K$ or less.

(Reference Paper)
https://www.ams.org/journals/tran/1993-339-02/S0002-9947-1993-1113693-1/S0002-9947-1993-1113693-1.pdf

This seems like it is close to the result I am looking for. Earlier in the paper, it is also stated that there are no conjugate points in $N(p,fracpiK)$.

Is there a simple step from this result that proves that the injectivity radius is nonzero?

Yes, they show that any compact Riemannian manifold with boundary is locally $mathrmCAT(kappa)$ for some $kappainmathbbR$.
In particular the injectivity radius is positive.