Suppose $S\subset \mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to\mathbb{R}^2$ is a continuous map with $\Gamma(0)=\Gamma(1)$. Suppose $\Gamma$ passes through all extreme points of $S$. (Then the convex hull of $\Gamma([0,1])$ contains $S$ by the Krein-Milman Theorem.) Let $|\Gamma|$ denote the arc length of $\Gamma$ defined in the usual way.
For a given such $S$, what is the smallest possible value of $|\Gamma|$?
Let $C(S)$ is the circumference (perimeter) of $S$.
Is it true that for all $S$ and $\Gamma$ satisfying the above conditions, $|\Gamma|\ge C(S)$?
Obviously, there is a $\Gamma_0:[0,1]\to \mathbb{R}^2$ with $|\Gamma_0|=C(S)$, so if this last question is answered affirmatively, then we have found the minimum length curve.
Simplified Question: If $S$ has finitely many extreme points, it's a convex polygon. In this case, the question reduces to: Is the shortest cycle visiting all vertices of a convex polygon its perimeter path?
Comment: This seems like a really fundamental and basic question that surely has been asked and answered and would appear as a basic theorem in textbooks on related subjects. But I can't find its answer.