Suppose $(M,g)$ is a compact Riemannian manifold with smooth boundary. We call a point $p \in M$ regular if there exists a geodesic of finite arc length passing through $p$ with end points on $\partial M$ (By end points I mean two points on the geodesic which also lie on $\partial M$ and such that the geodesic hits the boundary non-tangentially at these points).

(a) Is it true that the set of points $p \in M$ which are not regular has measure zero?

If the answer to part (a) is affirmative I would also like to pose a second part to question as follows:

Given any regular point $p \in M$, we consider the set of all geodesics of the above type passing through $p$. We say $p$ is optimal if there is no conjugate points to $p$ on at least one of these geodesics.

(b) Is it true that the set of non-optimal points in $M$ have measure zero?

Thanks, Thanks,

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