This problem seeks a relationship between the lengths of geodesics which emanate from a point on a surface in $\mathbb{R}^3$ and then come together again, and the bounds on the lengths of centred normal fibres through them in a maximal tubular neighbourhood of the surface.

Let $S \subset \mathbb{R}^3$ be a smooth closed (compact without boundary) surface. The local reach is a function $\rho: S \rightarrow \mathbb{R}$ such that $\rho(x)$ is the radius of the smaller of the two maximal empty tangential balls on either side of $S$ at $x$. An empty tangential ball at $x$ is an open Euclidean ball in $B \subset \mathbb{R}^3$ that does not intersect $S$, and whose boundary sphere is tangent to $S$ at $x$. It is maximal if it is not contained in any other such ball.

Let $\alpha$ and $\gamma$ be two geodesics which emanate from $p \in S$ in distinct directions and then meet again at $q \in S$.

I wish to show that there exists some point $z$ on either $\alpha$ or $\gamma$ such that $$ > \rho(z) \leq \frac{\ell(\alpha) + \ell(\gamma)}{2\pi}, > $$ where $\ell(\alpha)$ is the length of $\alpha$.

If for example $\alpha$ were the shorter geodesic and it contained a point conjugate to $p$, then our bound would follow from a standard result that relates the length of $\alpha$ to a bound on the square root of the sectional curvature it sees (the local reach is bounded by the radii of the osculating balls).

Also, if $\alpha$ and $\gamma$ together form a geodesic loop, then the result follows by considering the curvature of that loop as a space curve ( ... I asked for help with that too).

These two observations allow us to show that at least somewhere within a geodesic disk centred at $p$ and of radius bounded by the injectivity radius at $p$, we can find a $z$ satisfying the inequality, but I want more. I want to find such a $z$ on one of these two arbitrary geodesics.

This question came up within the context of surface sampling, where a popular means of defining the sampling density is through a function that is bounded by the local reach. We worked around it, but I'm still curious, and hope it might provide insight.


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