Geometry of Curves and Surfaces — Problem 5.3

Definitions and mathematical statement

Let $\gamma:S^1\to\mathbb R^3$ be a closed rectifiable curve of length $L>0$, and let $K=\operatorname{conv}(\gamma(S^1))$. For a full-dimensional hull, let $A(K)$ be its boundary surface area. Extend this quantity continuously to flat hulls: a planar convex region has area counted twice, while a segment has area zero. Equivalently, with $B$ the unit ball, \[ A(K)=\lim_{\varepsilon\downarrow0} \frac{V(K+\varepsilon B)-V(K)}{\varepsilon}. \] Here $+$ denotes Minkowski addition and $V$ is three-dimensional volume.

The conjecture is \[ A(K)\le\frac{L^2}{2\pi}. \] A planar circle of radius $L/(2\pi)$ attains the proposed value: its hull is a disk counted on both sides. Determine also the equality cases.

The curve is not required to be planar, simple, or contained in the boundary of its convex hull; repeated traversal contributes to $L$. In particular, a proof valid only when the curve splits the hull boundary into two spanning disks would cover a restricted case. The doubled-area convention for flat hulls is essential to the stated constant and its extremal example.

Short English statement

Does a planar circle maximize the surface area of the convex hull among all closed space curves of the same length, counting a flat hull on both sides?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 7000022 (AMR-069-0022), 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 5.3, PDF page 14. Original source indexed by AMR-069-0022. https://people.math.gatech.edu/~ghomi/Papers/op.pdf.