Dimension-free cotype for isotropic log-concave random polytope spaces

Abstract: Let $X_1,\ldots,X_N$ be independent random vectors in $\mathbb{R}<sup>n$ with common isotropic log-concave distribution $μ$ and set $$P_{N,n}^{μ:=\operatorname{conv}{\pm} X_i:1\leqslant i\leqslant N}$$. Assume that $N/n=γ\geqslant γ<em>0$ where $γ_0&gt;1$ is an absolute constant. We prove that with probability at least $1-Cγ\exp(-c n<sup>{1/4})$ every $k$-dimensional subspace $E$ of $$(\mathbb{R}^n,|\cdot|</em>{P_{N,n}<sup>μ})$$ satisfies $$d_{\mathrm{BM}} (E,\ell_\infty^k) \geqslant cγ<sup>{-C}k<sup>α$$ for every $1\leqslant k\leqslant n$ where $c,C,α&gt;0$ are absolute constants. Consequently, with the same probability, $(\mathbb{R}<sup>n,|\cdot|<em>{P</em>{N,n}<sup>μ})$ has cotype $q(γ)&lt;\infty$ with cotype constant depending only on $γ$, in particular the cotype exponent and the cotype constant are independent of $n$ and of $μ$. The proof adapts the deterministic coefficient scheme of Huang-Tikhomirov replacing the Gaussian estimates in their argument by estimates for isotropic log-concave random matrices. As an application, using the log-concave extension of Gluskin's theorem, we obtain a separable Banach space of finite cotype for which the Banach-Mazur diameter of its $k$-dimensional subspaces is of order $k$ and whose finite-dimensional building blocks are generated by isotropic log-concave random polytopes.

• The paper establishes dimension-free cotype estimates for Banach spaces from isotropic log-concave random vectors, generalizing Gaussian results to broader measures.
• It employs advanced log-concave matrix deviation techniques and combinatorial arguments to derive explicit high-probability bounds independent of dimensionality.
• The results provide new insights into random convex geometry and Banach space structures, impacting high-dimensional analysis and algorithmic applications.

Dimension-Free Cotype for Isotropic Log-Concave Random Polytope Spaces

Introduction and Motivation

This work establishes dimension-free cotype estimates for normed spaces constructed from isotropic log-concave random vectors in $\mathbb{R}^n$, generalizing prior results in the Gaussian regime to arbitrary log-concave measures. Specifically, for $N$ independent $\mu$-distributed isotropic log-concave random vectors $X_1,\ldots,X_N$ in $\mathbb{R}^n$, the study concerns the Banach space $(\mathbb{R}^n, \|\cdot\|_{P_{N,n}^\mu})$, where $$P_{N,n}^\mu = \mathrm{conv}\{\pm X_i : 1\leq i \leq N\}$$, focusing on the cotype properties of the associated norm.

The main innovation is the extension of dimension-free, quantitative cotype bounds (previously for the standard Gaussian case) to the substantially larger universality class of log-concave measures without symmetry, product, or rotational structure, and with explicit control over how the cotype behavior depends on the aspect ratio $\gamma = N/n$.

Main Results

The author proves that for any fixed $\gamma$ above a universal constant and for all $n$ large, with high probability over the random choice of the $N$0, the space $N$1 has cotype $N$2 with constant depending only on $N$3 and not on $N$4 or the measure $N$5. Concretely, the main structural and probabilistic estimates are as follows:

• Quantitative subspace exclusion: For all $N$6, with high probability, every $N$7-dimensional subspace $N$8 of $N$9 satisfies

$\mu$0

for universal $\mu$1, i.e., $\mu$2 is at least polynomially far (in $\mu$3) in Banach--Mazur distance from $\mu$4.

• Dimension-free cotype: The main consequence, using the Maurey–Pisier theorem, is that the entire space has cotype $\mu$5 with constant $\mu$6, both independent of $\mu$7.
• Banach--Mazur diameter of subspaces: For certain choices of the parameters, one obtains explicit randomized Banach spaces (from log-concave polytopes) where the diameter function $\mu$8 for large $\mu$9, matching the extremal Gluskin-type behavior even in finite cotype Banach spaces.

Notably, these theorems hold uniformly over all isotropic log-concave measures---the only requirement on $X_1,\ldots,X_N$0 is isotropy (mean zero, identity covariance) and log-concavity.

Methodology

The proof adapts the deterministic coefficient scheme of Huang–Tikhomirov, which originally handled the Gaussian case, and extends it to the log-concave setting. Key technical innovations include:

• Log-concave matrix deviation estimates: All Gaussian-specific arguments are replaced by the sharpest available restricted isometry and norm concentration results for isotropic log-concave ensembles (see e.g., [ALPTJ10], [ALPTJ11], [Guédon--Milman 2011]).
• Deterministic combinatorial argument: The scheme organizes a contradiction by assuming the presence of almost isometric $X_1,\ldots,X_N$1 subspaces, tracing the “large coordinate” structure of minimal $X_1,\ldots,X_N$2 representations with respect to the polytope norm, and exploiting the log-concave structure to push through a lower bound.
• Sparse structure exploitation: Concentration is quantified for both sparse and full random matrices; the log-concave norm deviation and restricted isometry properties are exploited to control all relevant events needed for the deterministic scheme.
• Banach space reductions: Quantitative applications of the Maurey–Pisier cotype theorem bridge exclusion estimates for $X_1,\ldots,X_N$3 subspaces to global cotype conclusions.

Key Quantitative and Structural Findings

• The lower bound on Banach--Mazur distance between any $X_1,\ldots,X_N$4-dimensional subspace and $X_1,\ldots,X_N$5 improves upon the trivial bound provided $X_1,\ldots,X_N$6, and is nontrivial throughout the regime $X_1,\ldots,X_N$7 for moderate aspect ratios.
• The cotype exponent and constant are both shown to be functionally independent of $X_1,\ldots,X_N$8 and the ambient measure $X_1,\ldots,X_N$9, depending only (explicitly and polynomially) on the aspect ratio parameter $\mathbb{R}^n$0.
• The probability bounds are exponentially strong: the main exclusion estimates and cotype properties hold with probability at least $\mathbb{R}^n$1.

Comparison with Prior Work

Compared to the Gaussian-specific theory, most notably Huang–Tikhomirov (Huang et al., 5 Mar 2026), the present results achieve:

• Universality across all isotropic log-concave measures, with no reliance on symmetry or product structure.
• Slightly weaker (polynomial) dependence on the aspect ratio parameter $\mathbb{R}^n$2 in loss constants.
• An explicit high-probability bound stronger than previous $\mathbb{R}^n$3 probabilities in the Gaussian regime, at the expense of a weaker (sub-Gaussian) exponent $\mathbb{R}^n$4 in the dimension.

Applications

An immediate application, via the log-concave extension of Gluskin’s theorem, is the existence of Banach spaces of finite cotype with maximal Banach–Mazur diameter growth, where the building blocks are constructed from random polytopes defined by arbitrary isotropic log-concave measures. This demonstrates a new universality phenomenon for local Banach space structure among finite cotype spaces, previously thought possible only in spaces containing $\mathbb{R}^n$5 uniformly.

Theoretical and Practical Implications

Theoretical

• The results demonstrate that key structural and metric properties associated with high-dimensional random convex bodies, such as cotype and Banach–Mazur distance scaling, are universal across all isotropic log-concave ensembles, not just the Gaussian case.
• These findings reinforce the robustness of log-concave universality in asymptotic/local convex geometry and high-dimensional Banach space theory.
• The methodology for dimension reduction, anti-concentration, and isometric embedding exclusion can likely inform further structural investigations for random normed spaces generated from broad high-entropy ensembles.

Practical

• The findings have implications for random geometric constructions in the study of normed space embeddings, functional analysis, and potentially sampling-based convex optimization methods.
• The probabilistic bounds for isomorphic subspace structure in random polytope spaces feed into average-case analysis for algorithms involving random matrices and convex polytopes.

Possible Future Directions

• Localization and refinement: Extending the approach to more refined local geometric invariants (e.g., type, VC-dimension, entropy numbers) for log-concave random Banach spaces.
• Dependent and structured ensembles: Considering whether certain weak dependence or deviation from isotropy can be handled with suitably modified techniques.
• Algorithmic consequences: Exploring connections to randomized algorithms in high-dimensional geometry and data science contexts, specifically where log-concave sampling is present.

Conclusion

This work substantively extends the geometric and probabilistic analysis of random Banach spaces generated by polytopes from isotropic log-concave distributions. It proves that sharp dimension-free cotype and subspace projection results are universal across this broad class, not restricted to the Gaussian setting. The explicit quantitative estimates and the associated combinatorial-deterministic methodology open the way to further universality results for random convex geometry and the local theory of Banach spaces.

Reference: "Dimension-free cotype for isotropic log-concave random polytope spaces" (2607.10373)