# Differential geometry: non-uniqueness of the closest point on a curve

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.