TITLE: Largest possible volume of the convex hull of a curve of unit length
QUESTION [26 upvotes]: What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?

REPLY [7 votes]: Here is an image of the optimal open convex curve.
Taken from Open Problems from CCCG 2012,
based on this paper, which cites Nudel'man (1975):

Paolo Tilli. 
  "Isoperimetric inequalities for convex hulls and related questions."
  Trans. Amer. Math. Soc. 362 (2010), 4497-4509.