TITLE: Largest convex hull of a unit length path
QUESTION [8 upvotes]: What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?

REPLY [15 votes]: The answer seems to be $\frac{1}{2\pi}$, using a semi circle. See 
Moran, P. A. P. "On a problem of S. Ulam." Journal of the London Mathematical Society 1.3 (1946): 175-179.