Banach-Mazur distances and basis constants of isotropic log-concave random spaces

Abstract: We study the Banach-Mazur distance between random normed spaces generated by centrally symmetric random polytopes associated with isotropic log-concave measures in $\mathbb{R}^n$. We show that, in a wide range of parameters, if $x_1,\dots,x_m$ and $y_1,\dots,y_m$ are independent samples from an isotropic log-concave probability measure on $\mathbb{R}^n$, then the corresponding normed spaces $X_{B_m}$ and $Y_{A_m}$ generated by their absolute convex hulls satisfy, with high probability, $$d_{\rm BM}(X_{B_m},Y_{A_m}) \geqslant \frac{cn}{\ln(1+m/n)},$$ which is sharp in both $n$ and $m$ and recovers the extremal order $n$ when $m \approx n$. Our results extend Gluskin's theorem from the Gaussian setting to general isotropic log-concave measures, providing evidence for a universality phenomenon in the extremal geometry of the Banach-Mazur compactum. In addition, we investigate operator-theoretic properties of the associated random spaces and, as consequences, we derive sharp estimates for their basis constant and show that these random spaces are far from the class of spaces with a $1$-unconditional basis. The proofs combine probabilistic and geometric methods with recent advances related to Bourgain's slicing problem.

• The paper establishes sharp asymptotic bounds for Banach–Mazur distances using random polytopes from isotropic log-concave measures.
• It demonstrates that the maximal separation scales optimally with dimension, extending Gluskin’s Gaussian results to a broader framework.
• The study leverages volumetric methods, concentration inequalities, and solutions to Bourgain’s slicing problem to analyze basis constants and operator norms.

Banach–Mazur Distances and Basis Constants in Isotropic Log-Concave Random Spaces

Introduction and Theoretical Context

The paper establishes sharp asymptotic estimates for Banach–Mazur distances between high-dimensional normed spaces generated by centrally symmetric random polytopes whose vertices are sampled from arbitrary isotropic log-concave probability measures on $\mathbb{R}^n$. This extends the classical theory initiated by Gluskin, who computed the extremal order of the Banach–Mazur compactum via random spaces constructed from Gaussian or spherical vectors, to a significantly broader and less symmetric probabilistic framework.

A central theme is the universality of extremal distances in the Banach–Mazur geometry: the work demonstrates that the maximal separation phenomenon in the Banach–Mazur metric, known from the Gaussian setting, is retained for polytopes generated from isotropic log-concave measures, modulo optimal logarithmic corrections. The results leverage recent progress on Bourgain's slicing problem, which asserts the boundedness of isotropic constants for log-concave measures, to circumvent the lack of invariance and product structures characteristic of more restrictive ensembles.

Main Results

Banach–Mazur Distance Estimates

Let $x_1,\ldots,x_m$ and $y_1,\ldots,y_m$ be independent samples from an isotropic log-concave probability measure in $\mathbb{R}^n$. Two models for random polytopes are considered:

• Basis-enriched model: $$B_m = \operatorname{absconv}\{e_1,\ldots,e_n, x_1/\sqrt{n},\ldots, x_m/\sqrt{n}\}$$
• Pure random polytope model: $B_m = \operatorname{absconv}\{ x_1,\ldots,x_m \}$

Let $X_{B_m}$ (resp.\ $Y_{A_m}$) denote the normed spaces with unit balls $B_m$ (resp.\ $A_m$). The main theorem asserts that, with high probability as $x_1,\ldots,x_m$0, uniformly over permissible $x_1,\ldots,x_m$1 (in a suitable range up to exponential in $x_1,\ldots,x_m$2),

$x_1,\ldots,x_m$3

where $x_1,\ldots,x_m$4 is the Banach–Mazur distance, and the dependence on both $x_1,\ldots,x_m$5 and $x_1,\ldots,x_m$6 is optimal. In the regime $x_1,\ldots,x_m$7, the lower bound achieves $x_1,\ldots,x_m$8, thus extending Gluskin's extremal construction from Gaussian to general isotropic log-concave frameworks.

Operator-Theoretic and Geometric Properties

The authors establish that random spaces of this type generically possess large basis constants and are far from possessing a 1-unconditional basis. In particular, for $x_1,\ldots,x_m$9 in the pure random polytope model, with high probability,

$y_1,\ldots,y_m$0

where $y_1,\ldots,y_m$1 denotes the basis constant of the space. This directly implies that random spaces of this type are maximally distant—of order $y_1,\ldots,y_m$2—from spaces admitting a 1-unconditional basis when $y_1,\ldots,y_m$3.

Furthermore, for any projection $y_1,\ldots,y_m$4 of rank between $y_1,\ldots,y_m$5 and $y_1,\ldots,y_m$6 one has

$y_1,\ldots,y_m$7

with overwhelming probability. This quantifies the operator-norm growth of projections in high-dimensional random spaces.

Universality and Comparison with Prior Work

Gluskin’s random space construction served to show that the diameter of the Banach–Mazur compactum is of order $y_1,\ldots,y_m$8, using Gaussian randomness and leveraging rotational invariance. The results here significantly generalize this extremal behavior. By extending to isotropic log-concave measures—where independence and rotational invariance are absent—this work demonstrates a robust universality, driven by deep measure concentration phenomena and volumetric estimates available due to the solution to Bourgain’s slicing problem. Notably, the latter provides uniform control on isotropic constants, eliminating a major technical barrier.

Methodological Innovations and Proof Techniques

Key techniques involve:

• Volumetric and entropy methods tailored to polytopes generated from log-concave vectors, with sharp control of volumes and covering numbers in high dimension.
• Probabilistic concentration inequalities for isotropic log-concave measures, including large deviation and small ball probability bounds (Paouris’ inequalities).
• Utilization of the slicing problem solution (by Klartag, Lehec, Guan) to circumvent issues stemming from lack of invariance or product structure.
• Extensions of operator-theoretic frameworks for mixing operators, following Szarek and Mankiewicz-Tomczak-Jaegermann, generalized to the isotropic log-concave regime.
• Careful combinatorial and approximation arguments (nets, covering numbers) to handle the randomness and weak dependence among the facets and vertices of the random polytopes.
• Exploitation of duality and convex geometric inequalities (e.g., John's ellipsoid, Blaschke–Santaló, Bourgain–Milman).

Implications and Future Directions

The demonstration of universality of maximal Banach–Mazur distances for isotropic log-concave ensembles has broad implications in the local theory of Banach spaces and in high-dimensional convex geometry. Specifically, it indicates that extremal metric properties of the Banach–Mazur compactum are governed by general measure concentration phenomena rather than specific features of Gaussian or product-type measures.

From a practical standpoint, these results inform the geometry of random constructions in functional analysis and potentially in high-dimensional data analysis, where random projections and polytopes generated from log-concave distributions play a role.

On the theoretical side, future work may examine:

• Finer estimates for the shape and spectral properties of random polytopes generated by even more general measures.
• Potential universality of other geometric constants, perhaps in non-centrally symmetric or non-isotropic settings.
• Applications to randomness in high-dimensional algorithms, random matrix theory, and geometric functional analysis.

Conclusion

This work establishes optimal, sharp-order bounds for Banach–Mazur distances and basis constants of normed spaces defined as the Minkowski compactum of random polytopes from isotropic log-concave measures, significantly generalizing Gluskin's classical constructions. By leveraging recent advances in the local theory of normed spaces and probabilistic convex geometry—notably the solution to Bourgain's slicing problem—the authors demonstrate that extremal geometric phenomena in Banach spaces are far more universal than previously understood, transcending the Gaussian context and reflecting deep, measure-theoretic structures inherent in high dimensions.