A new proof of the description of the convex hull of space curves with totally positive torsion

Abstract

We give new proofs of the description convex hulls of space curves ${\hat{I}}^3:[a,b]\mapsto {\mathbb{R}}^d$ having totally positive torsion. These are curves such that all the leading principal minors of $d\times d$ matrix $(γ', γ'', \ldots, γ^{(d)})$ are positive. In particular, we recover parametric representation of the boundary of the convex hull, different formulas for its surface area and the volume of the convex hull, and the solution to a general moment problem corresponding to ${\hat{I}}^3$.

36 pages, 10 figures. The main results of the paper follow (or are included) in two monographs M.G. Krein and A. A. Nudelman, The Markov Moment Problem and Extremal Problems; and S. Karlin and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics. In our paper we give a new self contained proofs of these results