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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?

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    $\begingroup$ By "largest", do you mean largest area? $\endgroup$
    – Mark Meckes
    Commented Sep 9, 2013 at 9:32
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    $\begingroup$ Yes, the convex hull having the largest area. $\endgroup$
    – ARi
    Commented Sep 9, 2013 at 9:47
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    $\begingroup$ The same question seems even more intriguing in dimension $3$. $\endgroup$
    – Benoît Kloeckner
    Commented Sep 9, 2013 at 21:05
  • $\begingroup$ It may just be an open problem in Dimension 3, see here on MO. $\endgroup$
    – ARi
    Commented Sep 11, 2013 at 15:04

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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.

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    $\begingroup$ "Seems" seems to be an understatement. Moran also proves that the semicircle is the unique maximizer. $\endgroup$
    – Misha
    Commented Sep 9, 2013 at 10:43
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    $\begingroup$ @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
    Commented Sep 9, 2013 at 10:59

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