Let P be a convex polytope in Euclidean space $\mathbf{R}^3$, i.e., the convex hull of a finite number of points. Let $\Gamma$ be a closed curve which passes through all vertices of $P$. How small can the length $L$ of $\Gamma$ be? More specifically, if $A$ denotes the area of the boundary of $P$, can one show that $$ L^2\geq 2\pi A $$ with equality only in the limiting case where $\Gamma$ converges to a circle in the plane, and $P$ becomes degenerate as its boundary collapses to a doubled disk.

Note that the shortest curve for a given polytope needs not lie on the boundary of $P$. For instance consider a short wide octahedron.

The above problem is the discrete version of a more general question which may be stated as follows. Among all closed curves of a given length in $\mathbf{R}^3$, which one maximizes the surface area of its convex hull? Of course if the above inequality is established for polygonal curves, then the general case follows.

The earliest reference to this problem seems to be in the book of Croft-Falconer-Guy, p. 38 (I would be interested to know if there is an earlier reference). A few years later it was discussed in a survey article of Zalgaller, who noted that the solution is easy if the curve lies on the boundary of the convex hull, in which case we might apply Weil's isoperimetric inequality for surfaces of nonpositive curvature.

I wanted to advertise the discrete version of the question here, because I think it should be more accessible, and it does not seem to be so well-known. The analogues problem for the volume of the convex hull is better known, and is also open.