Geometry of Curves and Surfaces — Problem 4.1

Definitions and mathematical statement

Let $S$ be the boundary of a compact convex body in $\mathbb R^3$, with its intrinsic length metric $d_S$. Here $d_S(x,y)$ is the infimum of lengths of curves joining $x$ to $y$ on the surface, not their Euclidean chord length. Write \[ D=\sup_{x,y\in S}d_S(x,y),\qquad A=\operatorname{area}(S). \] The extremal question asks whether every such surface satisfies $A\le\pi D^2/2$, with the sharp extremal metric given by a doubled Euclidean disk.

The doubled disk of radius $r$ is the intrinsic surface obtained by taking two copies of a flat closed disk and gluing their boundary circles pointwise. It has area $2\pi r^2$ and intrinsic diameter $2r$, explaining the proposed constant. It is an allowed degenerate convex-surface limit, not the ordinary single disk counted once.

If only boundaries of bodies with nonempty three-dimensional interior are admitted, interpret the question as the corresponding sharp supremum, approached by increasingly flattened bodies. A complete result should determine the equality or limiting case as well as the inequality. The question includes nonsmooth convex surfaces; restricting to a symmetric subclass gives only a special case.

Short English statement

Among convex surfaces of a fixed intrinsic diameter, is a doubled flat disk the area maximizer, giving the sharp bound $A\le\pi D^2/2$?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 7000017 (AMR-069-0017), 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 4.1, PDF page 12. Original source indexed by AMR-069-0017. https://people.math.gatech.edu/~ghomi/Papers/op.pdf.