This question stems from the discussion in:



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> [how to define the injectivity radius of manifolds with boundary?](https://mathoverflow.net/questions/236186/how-to-define-the-injectivity-radius-of-manifolds-with-boundary)
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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 $N(p,\frac{\pi}{2K})$ is open in M and the
distance function from p is convex on $N(p,\frac{\pi}{2K})$.


Where $N(p,\frac{\pi}{2K})$ is the set of points connected to $p$ by a unique geodesic of length $\frac{\pi}{2K}$ or less.



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> (Reference Paper)
>  <https://www.ams.org/journals/tran/1993-339-02/S0002-9947-1993-1113693-1/S0002-9947-1993-1113693-1.pdf>
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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 $N(p,\frac{\pi}{K})$.


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



Injectivity radius of manifolds with boundary
Yes, they show that any compact Riemannian manifold with boundary is locally $\mathrm{CAT}(\kappa)$ for some $\kappa\in\mathbb{R}$.
In particular the injectivity radius is positive.


