Geometry of Curves and Surfaces — Problem 5.2

Definitions and mathematical statement

A closed rectifiable space curve is a continuous map $\gamma:S^1\to\mathbb R^3$ of finite positive length. Length counts the traversal with multiplicity, so retracing an arc consumes length again. Let $K_\gamma$ be the compact convex hull of its image and let $V(K_\gamma)$ be its ordinary three-dimensional volume.

For $L>0$, determine \[ V_*(L)=\sup\{V(K_\gamma):\gamma\text{ is closed and rectifiable}, \ \operatorname{length}(\gamma)=L\}. \] Find the sharp value and characterize maximizing curves, or maximizing limiting configurations if the supremum is not attained in a more restrictive smooth class.

Scaling a curve by a factor $t$ multiplies length by $t$ and volume by $t^3$, so the main numerical target is the dimensionless constant $V_*(1)$, with $V_*(L)=L^3V_*(1)$. The curve may be nonplanar and need not lie on the boundary of its convex hull.

The closure condition is part of the admissible class. Results for open arcs, centrally symmetric curves, or curves with a specified knot type do not by themselves determine this general supremum. The volume being optimized belongs to the convex hull, not to a choice of spanning surface.

Short English statement

How much volume can the convex hull of a closed space curve enclose when the curve has a fixed total length? Find the sharp constant and extremal shapes.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 7000021 (AMR-069-0021), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ghomi - Open Problems in Geometry of Curves and Surfaces (2019), Problem 5.2, PDF page 13. Original source indexed by AMR-069-0021. https://people.math.gatech.edu/~ghomi/Papers/op.pdf.