Let $C$ be a finite-length, continuous 2D curve segment.
Given a 2D direction $d$, let the width of $C$ along $d$ be the length of the projection of $C$ on $d$. Or, equivalently, consider two parallel lines orthogonal to $d$: the width is the distance between the two farthest lines that both intersect $C$.
For a given curve segment $C$, let $m$ be the minimum value of the width of $C$ along all directions. For example, if $C$ is a straight segment, $m = 0$ because its width along the orthogonal direction is 0. If $C$ is a circumference, $m$ is its diameter.
I am interested in finding the curve $C$ with length 1 and the maximum possible value of $m$.
If we limit the search to closed curves, for symmetry the solution should be a circle with radius $\frac{1}{2\pi}$; then $m = \frac{1}{\pi}$.
If we allow $C$ to be an open curve then I feel that $m$ should increase, but I don't know where to look for the answer. My guess is that $C$ should be symmetric wrt the perpendicular bisector of the straight segment joining the two endpoints. I also suspect that its width should be the same regardless on $d$, and that convex hulls may be somewhat related.