# Largest convex hull of a unit length path

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?

• By "largest", do you mean largest area? – Mark Meckes Sep 9 '13 at 9:32
• Yes, the convex hull having the largest area. – ARi Sep 9 '13 at 9:47
• The same question seems even more intriguing in dimension $3$. – Benoît Kloeckner Sep 9 '13 at 21:05
• It may just be an open problem in Dimension 3, see here on MO. – ARi Sep 11 '13 at 15:04

The answer seems to be $\frac{1}{2\pi}$, using a semi circle. See