The convex hull of a convex space curve with four vertices

TL;DR

This paper establishes an upper volume bound for the convex hull of a special class of space curves with four vertices and characterizes when this bound is achieved, connecting classical geometric conditions with modern convex analysis.

Contribution

It provides a new upper bound for the volume of convex hulls of four-vertex curves and characterizes the extremal cases, linking historical geometric conditions with convex hull properties.

Findings

Upper bound for the volume of convex hulls of four-vertex curves.

Characterization of curves attaining the volume bound.

Connection to classical geometric conditions studied in the 1930s.

Abstract

We obtain an upper bound for the volume of the convex hull of a simple closed Frenet curve with exactly four vertices, i.e., four points of vanishing torsion, and lying on the boundary of its convex hull. Moreover, we show that the upper bound is attained when the curve intersects every plane in at most four points, a condition studied by Scherk and Segre in the 1930s. The proof relies on the fact that, under the four-vertex assumption, the convex hull is a union of line segments and therefore admits an elementary parametrization. We also comment on a question posed by Newson in 1899.