Geometry of Curves and Surfaces — Problem 4.2

Definitions and mathematical statement

Let $K\subset\mathbb R^3$ be a compact convex body with nonempty interior. Its support function is $h_K(u)=\max_{x\in K}\langle x,u\rangle$ for $u\in S^2$. The body has constant width $w>0$ when \[ h_K(u)+h_K(-u)=w\quad(u\in S^2). \] Thus every pair of parallel supporting planes is separated by the same distance. Write $A(K)$ for the area of its boundary and $V(K)$ for its three-dimensional volume.

For a prescribed $A_0>0$, determine \[ \inf\{V(K):K\text{ has constant width},\ A(K)=A_0\}, \] and characterize the bodies attaining the sharp bound, up to rigid motion. The numerical width is allowed to vary between competitors: the fixed quantity in this catalogue entry is surface area.

Equivalently, determine the minimum possible scale-invariant ratio $V(K)/A(K)^{3/2}$ among constant-width bodies. Nonsmooth convex boundaries are permitted, so minimizers are not ruled out merely because they contain seams or curvature jumps. The planar analogue optimizes area of constant-width curves; it does not itself settle this three-dimensional surface-area normalization.

Short English statement

Which three-dimensional body of constant width encloses the least volume when its surface area is prescribed?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 7000018 (AMR-069-0018), 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 4.2, PDF page 12. Original source indexed by AMR-069-0018. https://people.math.gatech.edu/~ghomi/Papers/op.pdf.