I try to understand: is there a smallest in area convex set that every smooth curve with length 1 can be placed inside it by translation and rotation?
I only have a upper bound $S \leq \frac14+\frac{\pi}{16}$ because of convex hull of two circles radius $\frac14$ and simple lower bound $S\geq\frac1{4\pi}$.
Does this set exist and what is its length?