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?
$\begingroup$
$\endgroup$

1$\begingroup$ By "largest", do you mean largest area? $\endgroup$ – Mark Meckes Sep 9 '13 at 9:32

1$\begingroup$ Yes, the convex hull having the largest area. $\endgroup$ – ARi Sep 9 '13 at 9:47

2$\begingroup$ The same question seems even more intriguing in dimension $3$. $\endgroup$ – Benoît Kloeckner Sep 9 '13 at 21:05

$\begingroup$ It may just be an open problem in Dimension 3, see here on MO. $\endgroup$ – ARi Sep 11 '13 at 15:04
$\begingroup$
$\endgroup$
The answer seems to be $\frac{1}{2\pi}$, using a semi circle. See

3$\begingroup$ "Seems" seems to be an understatement. Moran also proves that the semicircle is the unique maximizer. $\endgroup$ – Misha Sep 9 '13 at 10:43

1$\begingroup$ @Misha Yes. There are many variants of this question, for example taking an circle instead of an interval. See here for an overview. $\endgroup$ – Moritz Firsching Sep 9 '13 at 10:59