I have looked for theorems about the closest points, but I could not find such a theorem I need to establish some other claim. The main question is expressed as a proposition as follows:

Proposition: Consider a $C^2$ naturally parametrized curve $\boldsymbol{\gamma}:\mathbb{R} \rightarrow \mathbb{R}^3$ defined in the three-dimensional Euclidean space with $\Vert \boldsymbol{\gamma}''(s) \Vert > 0$ for all $s \in [s_0, s_1]$, $s_1 - s_0 \geq \epsilon > 0$. Then, there is a non-empty set $A \subset \mathbb{R}^3$ such that whenever ${\bf{p}} \in A$, ${\bf{p}}$ has at least two distinct closest points on $\boldsymbol{\gamma}$.

To help interpretation, $s$ is the arc length parameter, $\Vert \boldsymbol{\gamma}'(s) \Vert = 1$, $\kappa := \Vert \boldsymbol{\gamma}''(s) \Vert$ is the curvature of the curve.

Roughly speaking, the proposition states that if a twice continuously differentiable curve is bent on some interval, there is a region such that a point belongs to the region has always two or more closest points on the curve. A representative example is a straight line that passes through the center of a circle and is perpendicular to the plane the circle lies on. I want to show that there is such a point for more general curves. In contrast, a straight line, whose curvature is zero for the entire interval, can always have a unique closest point.

Is there any theorem related to this claim? Thank you.



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