not exist closed geodesics in ${\mathbb{R}}^n$, it is natural to look for geodesics contained in a bounded domain $\Omega \subset {\mathbb{R}}^n$ which meets $\partial \Omega$ orthogonally at its end points. These are called *orthogonal geodesic chords* (see Definition \[D:OGC\] for a precise definition). In [@Lusternik-Schnirelmann], Lusternik and Schnirelmann proved the following celebrated result:

\[T:LS\] Any bounded domain in $\mathbb{R}^n$ with smooth convex boundary contains at least $n$ distinct orthogonal geodesic chords.

Kuiper [@Kuiper64] showed that the same conclusion holds if the boundary is only $C^{1,1}$. For our convenience, we will assume that all the submanifolds and maps are $C^\infty$. Recall that the boundary of a domain $\Omega \subset {\mathbb{R}}^n$ is said to be (locally) *convex* if the second fundamental form $A$ of $\partial \Omega$ with respect to the unit normal $\nu$ (pointing into $\Omega$) is positive semi-definite, i.e. for all $p \in \partial \Omega$, $u \in T_p \partial \Omega$, we have $$\label{E:convex}
A(u,u):=\langle D_u u ,\nu \rangle  \geq 0,$$ where $D$ is the standard flat connection in ${\mathbb{R}}^n$. Notice that Theorem \[T:LS\] gives an optimal lower bound as seen in the example of the convex region bounded by the ellipsoid given by $$\Omega:=\left\{(x_1,\cdots,x_n) \in {\mathbb{R}}^n \; : \; \sum_{i=1}^n \frac{x_i^2}{a_i^2} 