Polytope Approximation in Surface-Area Deviation

Definitions and mathematical statement

A convex body in a finite-dimensional Euclidean space ${\mathbb{R}}^n$ is a compact convex set with nonempty interior. A convex polytope is the convex hull of finitely many points; its vertices are its extreme points. These definitions describe a standard Euclidean setting, but the supplied record does not specify the ambient dimension or expressly fix this setting.

The approximation question can be expressed conditionally as follows. Once an ambient space, an admissible class $\mathcal P(K)$ of approximating polytopes, and a deviation functional $D$ have been specified, define, for each positive integer $N$, \[ e_N(K;D,\mathcal P) = \inf\left\{D(K,P):P\in\mathcal P(K),\ \#\operatorname{vert}(P)\leq N\right\}, \] with the convention that the infimum of an empty set is $+\infty$. Here $\operatorname{vert}(P)$ denotes the vertex set of $P$.

Source question. Determine how well a convex body can be approximated by polytopes as a function of their number of vertices, particularly when the approximation error is measured by symmetric surface-area deviation. In a specified model, the corresponding target would be sharp bounds or asymptotic behavior for the optimal approximation error as the vertex budget tends to infinity.

Unresolved qualifications. The displayed infimum is a formulation template, not a fully specified conjecture extracted from the source. The record supplies no definition of symmetric surface-area deviation, no dimension range, no regularity or normalization assumptions on the body, and no conditions on containment or relative positioning of the approximating polytope. It also does not specify whether the vertex count is exact or an upper bound, whether estimates are pointwise in the body or uniform over a class, or what constitutes the desired sharp approximation result. These choices must be resolved before the question has a unique precise mathematical formulation.

Short English statement

The question asks for the best accuracy of polytope approximation in terms of the number of vertices, especially for symmetric surface-area deviation. The supplied record leaves the deviation functional and approximation model unspecified, so the formula above is conditional and requires human mathematical review.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30003065$, title Polytope Approximation in Surface-Area Deviation, and exact source problem code catalogue entry. The project identifies data/unsolvedmath/problems.json and the supplementary catalog/problems.json as its registries; their contents have not been independently inspected here.
• Supplied report attribution: Asymptotic Geometric Analysis (2016), Oberwolfach Reports, pp. 507--565, DOI 10.4171/owr/2016/11. https://doi.org/10.4171/owr/2016/11. The collection attributes the question to workshop open-problem discussions; no individual proposer is supplied.
• Supplied literature source labeled "surface-area deviation": https://doi.org/10.1007/s10959-016-0701-9. No further bibliographic details or precise supporting result are supplied.
• Supplied literature source labeled "polytopal approximation": https://doi.org/10.1017/9781009701761.022. No further bibliographic details or precise supporting result are supplied.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem partially solved and reports sharp asymptotic rates in many volume, Hausdorff, surface-area, and random-approximation models, while describing the broad question across arbitrary bodies, metrics, and positioning as model-dependent and incomplete. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.