Suppose $S\subset\mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to S$ is a continuous curve that passes through every extreme point of $S$, i.e., the convex hull of $\Gamma([0,1])$ is $S$. I am interested in obtaining a lower bound on the length $|\Gamma|$ of $\Gamma$.

I conjecture that $$ |\Gamma| \ge C(S) - \ell $$ where $C(S)$ is the perimeter (circumference) of $S$ and $\ell$ is defined as $$ \ell = \sup\{d(e,e'): e,e'\text{ are extreme points of $S$ and $\overline{ee'}\subset \partial S$} \}. $$ Here, $\overline{ee'}$ is the line segment joining $e$ and $e'$. Intuitively, $\ell$ is the length of the longest straight segment in $\partial S$. I think the shortest path $\Gamma$ with $\mathrm{ConvexHull}(\Gamma([0,1])) =S$ can be formed by traveling along the circumference of $S$ from one vertex of the longest straight edge in $\partial S$ to the other vertex. (Obviously, the direction of this path is the one giving $\mathrm{ConvexHull}(\Gamma([0,1])) =S$).

Is the conjectured bound on $|\Gamma|$ above correct? Can one construct a counterexample?

**Motivation** I'm trying to come up with a simple proof that Isbell's curve solves a particular instance of Bellman's "Lost in a Forest" problem. See this question on MSE if you want to know more about my motivation, but that background is not at all necessary to understand and solve this problem.