Georgia Institute of Technology College of Sciences

We show that in Euclidean 3-space any closed curve which contains the unit sphere in its convex hull has length at least $4\pi$, and characterize the case of equality, which settles a conjecture of Zalgaller. Furthermore, we establish an estimate for the higher dimensional version of this problem by Nazarov, which is sharp up to a multiplicative constant, and is based on Gaussian correlation inequality. Finally we discuss connections with sphere packing problems, and other optimization questions for convex hull of space curves. This is joint work with James Wenk.