Banach–Mazur Bounds and Mean Width of Inscribed Polytopes

Definitions and mathematical statement

Let $B^d=\{x\in{\mathbb{R}}^d:\|x\|_2\leq1\}$, let $d\geq1$, and let $1\leq k\leq d$. Let $P_{d+k}\subset B^d$ be a full-dimensional polytope with exactly $d+k$ vertices. "Inscribed" conventionally requires its vertices to lie on $S^{d-1}$, though the source also writes only containment and does not expand that convention.

Under a standard affine-sandwich normalization of multiplicative Banach--Mazur distance to the ball, write \[ \delta_{\mathrm{BM}}(P,B^d)=\inf\{\lambda\geq1:\exists A\text{ an invertible affine map},\ B^d\subseteq A(P)\subseteq\lambda B^d\}. \] The exact affine/translation convention must be checked for these nonsymmetric bodies; it is not a logarithmic distance or distance minus one. The first source conjecture is \[ \exists\widetilde c>0\quad\forall d\geq1\quad\forall k\in\{1,\ldots,d\} \quad\forall P_{d+k}\text{ as above},\qquad \delta_{\mathrm{BM}}(P_{d+k},B^d)\geq\widetilde c\,\frac d{\sqrt k}. \] The positive constant is absolute, independent of dimension, vertex count, and the polytope. No value for it is supplied.

For the second question, define the support function $h_P(u)=\max_{x\in P}\langle x,u\rangle$, and, with normalized spherical measure $\sigma$, the mean width \[ w(P)=\int_{S^{d-1}}\left(h_P(u)+h_P(-u)\right)\,{\,\mathrm{d}}\sigma(u). \] Let $\Delta_d$ be a regular simplex inscribed in $B^d$, whose unit vertices have pairwise inner product $-1/d$. Is it true that \[ \forall P_{d+1}\text{ inscribed},\qquad w(P_{d+1})\leq w(\Delta_d)? \] The question asks maximality, not an equality-case classification or uniqueness theorem. The spherical-measure normalization multiplies all mean widths by a common factor and does not change the comparison. The two questions are separate; no implication from the Banach--Mazur bound to the width maximum is asserted.

The supplied literature triage checked 2026-08-21 reports no authoritative complete resolution of this extracted pair of questions, without separate theorem statements or normalization details.

Short English statement

Seek the universal $d/\sqrt k$ lower bound on distance from a $(d+k)$-vertex inscribed polytope to the ball, and separately ask whether the regular simplex maximizes mean width among inscribed simplices. The nonsymmetric distance and inscription conventions need confirmation.

Sources

• Local UnsolvedMath registry, permanent ID 30000634, catalogue entry, Banach--Mazur Bounds and Mean Width of Inscribed Polytopes.
• Konvexgeometrie (2007), Oberwolfach Reports, pp. 3321--3396, DOI https://doi.org/10.4171/owr/2006/56.
• Supplied literature triage checked 2026-08-21 cites only that report; no separate bound or mean-width theorem citation is supplied.