The convex hull of a convex space curve with four vertices

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.