submanifold in ${\mathbb{R}}^n$ and hence proved the following result.

\[T:double-normals\] Any $k$-dimensional compact submanifold $\Sigma$ in ${\mathbb{R}}^n$ admits at least $k+1$ orthogonal geodesic chords.

Note that Theorem \[T:double-normals\] generalizes Theorem \[T:LS\] by taking $\Sigma$ to be the boundary of a bounded convex domain. However, we emphasize that if $\Sigma=\partial \Omega$ is the boundary of a non-convex domain $\Omega \subset {\mathbb{R}}^n$, then the orthogonal geodesic chords obtained in Theorem \[T:double-normals\] are not necessarily contained in $\Omega$ (recall Figure \[F:Bos\]).

The original proof of Theorem \[T:LS\], \[T:Bos\] and \[T:double-normals\] all used a discrete curve shortening process similar to the one introduced by Birkhoff [@Birkhoff17] in the study of existence of closed geodesics in Riemannian manifolds. A description of the process can be found in [@Gluck-Ziller83] (see also a modified version in [@Zhou16]). The curve shortening process, denoted by $\Psi$, take a piecewise smooth curve $c:[0,1] \to M$ with end points lying on $\Sigma$ to a piecewise geodesic curve $\Psi(c):[0,1] \to M$ which meets $\Sigma$ orthogonally at its end points. The most important properties of $\Psi$ are summarized below:

-   ${\operatorname{Length}}(\Psi(c)) \leq {\operatorname{Length}}(c)$ and equality holds if and only if $c$ is an orthogonal geodesic chord, in which case $\Psi(c)=c$.

-   $\Psi(c)$ depends 