Approximation of Smooth Convex Bodies by Inscribed Polytopes

Definitions and mathematical statement

Let $K\subset{\mathbb{R}}^d$ be a smooth convex body. For $N\geq d+1$, let $\mathcal P_N(K)$ be the class of full-dimensional polytopes $P\subseteq K$ with at most $N$ vertices, all lying on $\partial K$, using the usual inscribed-polytope convention.

One affine-invariant comparison is Banach--Mazur distance. For full-dimensional convex bodies $K,L$, an explicit general-body convention is \[ d_{\mathrm{BM}}(K,L)=\inf\{\lambda\geq1:\ K-a\subseteq T(L-b)\subseteq\lambda(K-a),\ a\in\operatorname{int}K,\ b\in\operatorname{int}L,\ T\in\operatorname{GL}_d({\mathbb{R}})\}. \] Corresponding approximation errors include \[ \begin{aligned} E_N^{\mathrm{BM}}(K)&=\inf_{P\in\mathcal P_N(K)}(d_{\mathrm{BM}}(K,P)-1),\\ E_N^{\mathrm{vol}}(K)&=\inf_{P\in\mathcal P_N(K)}(\operatorname{vol}_d(K)-\operatorname{vol}_d(P)). \end{aligned} \] The source asks how well approximation is possible under the complexity restriction, especially prescribed vertex number: determine optimal quantitative bounds or asymptotics for such errors in the intended smoothness and dimension classes. Other intended complexity measures must be specified separately.

Raw volume difference, although explicitly listed in the source, is not invariant under every invertible affine transformation: it scales by the absolute determinant of the linear part. Relative volume deficit $E_N^{\mathrm{vol}}(K)/\operatorname{vol}_d(K)$ is affine invariant. Both are distinguished rather than silently normalizing the source's volume difference. The extract does not specify differentiability order, positive curvature, fixed-body versus uniform estimates, exact versus at-most vertex number, or a requested rate, constant or asymptotic regime.

Short English statement

Determine optimal approximation of smooth convex bodies by inscribed polytopes with controlled complexity, especially vertex count. Banach--Mazur error and volume deficit are distinct targets, and only a normalized volume deficit is affine invariant under arbitrary affine changes.

Sources

• Supplied report: Konvexgeometrie (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/56, pp. 3321--3396; proposed year 2006. https://doi.org/10.4171/owr/2006/56.
• Local UnsolvedMath registry, permanent ID 30000633, catalogue entry; created 25 August 2026, updated 16 September 2026. The corrected_verified note says only ambient-space notation was changed, retaining restrictions and metrics. Its supplied 21 August 2026 literature triage labels the extracted direction open and reports no located authoritative complete resolution. No further external reference is supplied; the dated assessment is not independently updated here.