Geometry of Curves and Surfaces — Problem 4.3

Definitions and mathematical statement

Let $S\subset{\mathbb{R}}^3$ be the source's closed surface, with Euclidean diameter \[ d=\sup\{|x-y|:x,y\in S\}. \] For a unit vector $u$ and $t\in{\mathbb{R}}$, put \[ H(u,t)=\{x:\langle u,x\rangle=t\},\qquad Z(u,t,h)=\{x:t\le\langle u,x\rangle\le t+h\}. \] A precise global-constancy reading of the hypothesis is that there exist a separation $h<d$ and a number $A$ such that, for every $u\in S^2$ and $t\in{\mathbb{R}}$, \[ S\cap H(u,t)\ne\varnothing,\quad S\cap H(u,t+h)\ne\varnothing \quad\Longrightarrow\quad \mathcal H^2\left(S\cap Z(u,t,h)\right)=A. \] Thus it is the area of the part of the surface in a slab, not area of a planar section of an enclosed body. Both planes are taken to meet $S$, and $A$ is independent of direction and position.

The question is whether this forces \[ S=\{x\in{\mathbb{R}}^3:|x-c|=r\} \] for some $c\in{\mathbb{R}}^3$ and $r>0$.

The source states only $h<d$, although a nondegenerate distance condition needs $h>0$; allowing $h=0$ would change the problem substantially. This missing strict-positivity condition is flagged rather than silently added. The regularity, connectedness, embeddedness and compactness meaning of "closed surface," area convention, and whether "constant" is global or only directionwise also require confirmation. No convexity assumption is stated or imposed.

Short English statement

If every width-$h$ slab whose two boundary planes meet a closed surface contains the same amount of surface area, must the surface be a round sphere? The fixed width should be checked for strict positivity, and the area is surface area in the strip, not cross-sectional area.

The supplied summary calls the specific strip-area problem unresolved and mentions Funk/Radon and section-rigidity themes. Its unqualified statement about equal areas of all planar sections supplies no precise theorem or hypotheses and is not used as a solution. The OPEN-TRIAGE label and 2019 survey problem are retained with current status marked for review.

Sources

• Local UnsolvedMath registry, permanent ID 7000019, source code catalogue entry, "Geometry of Curves and Surfaces --- Problem 4.3," proposed by Mohammad Ghomi, 2019. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Mohammad Ghomi, Open Problems in Geometry of Curves and Surfaces (2019), Problem 4.3, PDF page 12, https://people.math.gatech.edu/~ghomi/Papers/op.pdf; normalized pdf-text extraction, accessed 2026-07-29.
• The research summary's section-rigidity and Funk/Radon context has no complete primary reference or exact statement. Source-specific provenance: the author survey labels the item a problem; current status and statement-reuse terms NEEDS_REVIEW for the public author-hosted PDF, separately from the dataset licence.