Let $(M,g)$ be a closed Riemannian manifold and let $N$ be a closed embedded submanifold. A *tube* $T(N,r)$ of radius $r$ of $N$ is defined as the set of points of $M$ which can be reached by a geodesic path of length $< r$ starting from $N$ orthogonally. We further require that $T(N,r)$ is open in $M$ and that the map $(p,v)\mapsto \exp_{p}(v)$ defined on $\{(p,v)\in TN^{\perp}\,\,:\,\, g_p(v,v)<r^2\}$ is a diffeomorphism onto $T(N,r)$.
Then the *normal injectivity radius* of $N$ (which we denote by ${\rm nir}(N)$) is defined as the maximal radius of a tube around $N$.

Assume that the sectional curvatures of $M$ and the second fundamental form of $N$ are uniformly bounded by a constant $K$. Is it possible to show that ${\rm nir}(N)\geq c(K)$, for some positive constant $c(K)$ depending only on $K$? Can this bound be obtained explicitly?