Let $M\subset R^n$ be a manifold with geodesic injectivity radius $r_x$ at $x$. Assume $r=\inf_{x\in M}r_x>0$. Is it possible to specify a $\delta>0$ such that for any $x,y\in M$ that are closer than $\delta$ in the embedding space ($\|x-y\|_2<\delta$), it holds that $d(x,y)<r$?
My guess is that this is true. If not, it would seem to me that $M$ is a fractal of sort.
Suppose it is false. Then there is a sequence $(y_n)_{n=1}^\infty$ such that $\lim_{n\rightarrow\infty}\|y_n-x\|=0$ but $\lim_{n\rightarrow\infty}d(y_n,x)\geq r$. This contradicts the continuity of $d$.