The convex hull of a convex space curve with four vertices
Jakob Bohr, Steen Markvorsen, Matteo Raffaelli
TL;DR
This work establishes a sharp upper bound on the volume of the convex hull of a simple closed convex curve with exactly four vertices in $\mathbb{R}^3$, showing $\mathrm{vol}(\mathrm{conv}(\gamma)) \le \frac{1}{24} \int\! \int \big| \det(\gamma'(t_{1}), \gamma'(t_{2}), \gamma(t_{2})-\gamma(t_{1})) \big| \; dt_{1}\, dt_{2}$, which implies $\mathrm{vol}(\mathrm{conv}(\gamma)) < \frac{1}{48} \mathrm{length}(\gamma)^{3}$ for closed curves. The authors exploit the four-vertex condition to prove the convex hull is a union of chords and to construct an elementary parametrization $\sigma(t_{1},t_{2},u)=\gamma(t_{1})+u(\gamma(t_{2})-\gamma(t_{1}))$, using a radial projection argument onto $S^{2}$ and antipodal-point analysis to derive the bound. They show that equality occurs when every plane intersects the curve in at most four points, linking the result to classical Scherk–Segre theory and discussing Newson's 1899 challenge, with a 3D tetrahedral-decomposition perspective providing a constructive path toward the area-like formula in three dimensions. Overall, the paper sharpens isoperimetric-type bounds for convex hull volumes of space curves and connects modern methods to classical geometric results.
Abstract
We obtain an upper bound for the volume of the convex hull of a simple closed Frenet curve having 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 hinges on the fact that, under the four-vertex assumption, the convex hull is a union of line segments, and so it admits an elementary parametrization. We also comment on a question posed by Newson in 1899.