when is a point on a curve that minimizes the distance to another given point (that is not on the curve) unique?

Question

If anybody has any useful references (or even the solution) for this problem, it would be much appreciated.

Suppose I have a smooth, closed-curve $S$ that lies in $\mathbb{R}^2$. Let $s \in S$ be a point on the curve. Given a point $P$ that lies inside $S$ (so $S$ encloses $P$), what are the conditions on $P$ (and possibly $S$) such that there is exactly one $s$ that minimizes the distance $|P-s|$?

In other words, if $d(P,s) = |P - s|$ (Euclidean distance) and $s \in S$ is treated as a variable with $P$ fixed, under what conditions does $d(P,s)$ have a unique global minimum with respect to $s$?

Note that I am not asking about uniqueness of the minimum distance, but the uniqueness of the minimizing point. Obviously, if $S$ is a circle and $P$ is its center, then there are infinitely many points on $s \in S$ that minimize the distance $d(P,s)$. The minimizing distance (the radius) is unique but the points are not. If $P$ is not the center, and still remains strictly inside the circle, the $s$ that minimizes $d(P,s)$ becomes unique.

Heuristically, I would expect a unique global minimum if $P$ lies close enough to $S$. For example, if it was within a radius of curvature, the unique, minimizing $s$ would be found by dropping a perpendicular.

Thanks in advance!

Answer 1

If $S$ was a circle and $P$ was in the center, every point of $S$ would be a minimizing point, even though it is within the radius of curvature, if that was, what you mean.

Answer 2

"when is the minimum distance from a point to a closed boundary unique?"

Always, of course.

(How could it possibly be otherwise?)

If your question is instead "when is the point on a curve closest to a given point not on the curve unique?" then there is no general answer. You must test every point (or make explicit model of the curve and the non-curve point.