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

TL;DR

This paper provides new proofs for the description of convex hulls of space curves with totally positive torsion, including parametric boundary representations, surface area formulas, volume calculations, and solutions to related moment problems.

Contribution

It introduces novel proofs and formulas for convex hulls of space curves with totally positive torsion, enhancing understanding of their geometric properties.

Findings

Parametric representation of convex hull boundaries

Formulas for surface area and volume of convex hulls

Solution to a general moment problem for such curves

Abstract

We give new proofs of the description convex hulls of space curves $\gamma : [a,b] \mapsto \mathbb{R}^{d}$ having totally positive torsion. These are curves such that all the leading principal minors of $d\times d$ matrix $(\gamma', \gamma'', \ldots, \gamma^{(d)})$ 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 $\gamma$.