Convex Hulls of Curves: Volumes and Signatures

TL;DR

This paper explores how path signatures can be used to compute the volume of the convex hull of certain curves, providing conditions and interpretations that connect stochastic analysis with computational geometry.

Contribution

It introduces sufficient conditions for using path signatures to determine convex hull volumes and proposes a conjecture for the necessary and sufficient conditions, linking geometric and algebraic properties.

Findings

Established conditions for signature-based volume computation

Identified classes of curves (cyclic, moment, totally positive torsion)

Provided geometric interpretation of the volume formula

Abstract

Taking the convex hull of a curve is a natural construction in computational geometry. On the other hand, path signatures, central in stochastic analysis, capture geometric properties of curves, although their exact interpretation for levels larger than two is not well understood. In this paper, we study the use of path signatures to compute the volume of the convex hull of a curve. We present sufficient conditions for a curve so that the volume of its convex hull can be computed by such formulae. The canonical example is the classical moment curve, and our class of curves, which we call cyclic, includes other known classes such as $d$-order curves and curves with totally positive torsion. We also conjecture a necessary and sufficient condition on curves for the signature volume formula to hold. Finally, we give a concrete geometric interpretation of the volume formula in terms of lengths and signed areas.