Dimension-free Gaussian tail estimates for linear functionals on convex bodies

Abstract: Let $K \subset \mathbb{R}^n$ be a centered convex body of volume one. We prove that there exist absolute constants $c,C > 0$ and an orthonormal set of vectors $Θ\subset S^{n-1}$ with size $\left|Θ\right| \ge 9n/10$ such that, if $X$ is a random vector uniformly distributed on $K$, then for all $θ\in Θ$ one has [ c\cdot \sqrt{p}\,\left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^2\right)^{1/2} \le \left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^p\right)^{1/p} \le C\cdot \sqrt{p}\,\left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^2\right)^{1/2}, ] where the upper estimate holds for all $p \ge 1$ while the lower bound only holds for $1 \le p \le n$.

• The paper establishes that nearly 9n/10 directions of any centered convex body have Gaussian tail behavior with absolute moment equivalence to the Gaussian case.
• The methodology blends Ψ2-body geometry, convex discretization, and Gaussian correlation techniques to achieve precise moment estimates.
• The results resolve Milman’s open problem, advancing high-dimensional convex geometry and paving new avenues in probabilistic analysis.

Dimension-Free Gaussian Tail Estimates for Linear Functionals on Convex Bodies

Introduction and Context

The study of one-dimensional marginals of high-dimensional convex bodies is central in contemporary convex geometric analysis, with profound connections to probability, asymptotic functional analysis, and high-dimensional statistics. A longstanding question, attributed to V. Milman, concerns the existence of a direction in which the marginal of the uniform distribution on an arbitrary centered convex body of volume one exhibits dimension-free subgaussian tail behavior. Previous advances provided such subgaussian directions in special cases and with dimension-dependent constants. However, the existence of a universal, dimension-free subgaussian constant for general convex bodies remained unresolved, representing a significant open problem at the interface of convex geometry and high-dimensional probability.

This work establishes a dimension-free Gaussian tail estimate for a large orthonormal family of directions on any centered convex body $K \subset \mathbb{R}^n$ of volume one. Specifically, there exists an explicit, large orthonormal set $\Theta \subset S^{n-1}$ with $|\Theta| \geq 9n/10$ such that for each direction $\theta \in \Theta$, the marginal $\langle X, \theta \rangle$ of a uniform vector $X$ on $K$ satisfies absolute moment equivalence to the Gaussian case up to absolute constants. Both upper and lower bounds of Gaussian order are proven, yielding strong tail implications. This result resolves Milman’s dimension-free subgaussian direction question in the affirmative and quantifies the prevalence of such directions.

Main Theorem and Its Implications

Theorem (Informal Statement)

Let $K \subset \mathbb{R}^n$ be centered, convex, $\operatorname{Vol}(K)=1$. There exist absolute constants $c, C > 0$ and an orthonormal set of directions $\Theta \subset S^{n-1}$0 with $\Theta \subset S^{n-1}$1 such that, for $\Theta \subset S^{n-1}$2 uniform on $\Theta \subset S^{n-1}$3 and all $\Theta \subset S^{n-1}$4, every $\Theta \subset S^{n-1}$5 satisfies: $\Theta \subset S^{n-1}$6 The upper inequality, with a possibly larger $\Theta \subset S^{n-1}$7, holds for all $\Theta \subset S^{n-1}$8.

This establishes dimension-independent, Gaussian-type moment growth and subgaussian tails for almost an entire orthonormal system of linear functionals. Standard implications, via Markov and Paley-Zygmund, yield two-sided Gaussian tail bounds for $\Theta \subset S^{n-1}$9.

This result is not restricted to random directions but produces explicit, almost full orthonormal systems, dramatically strengthening previous existential or probabilistic statements. For convex geometry and probability, this shows the subgaussian concentration phenomenon is not scarce but prevalent for linear functionals on high-dimensional convex bodies.

Technical Approach

Overview

The methodology blends geometric, probabilistic, and analytic tools, with key ingredients including:

• $|\Theta| \geq 9n/10$0-Bodies: The proof relies on the geometry of the $|\Theta| \geq 9n/10$1 centroid-type body, encoding the subgaussian behavior of linear marginals via the position of the polar in Euclidean space. Bounding moments via this convex body translates the probabilistic question to a geometric one.
• Convex Body Discretization and Gaussian Correlation: The defining constraints are discretized over dyadic scales, and geometric intersection estimates (using the Gaussian correlation inequality) establish the existence of directions simultaneously controlling all relevant moments. Unlike [GPV11], which used covering number arguments, the present work obtains sharper volumetric estimates directly in Gaussian space, exploiting recent small-ball probability technologies.
• Iterative Construction: By removing small sublevel sets and carefully selecting within their orthogonal complements, the construction extracts a large orthonormal system $|\Theta| \geq 9n/10$2 for which the desired inequalities hold uniformly.
• Negative Moment and Small Ball Estimates: Recent results on isotropic constants (notably the solution of Bourgain's slicing problem [KL25, (Bizeul, 12 Jan 2025)]) and sharp small-ball probability bounds underpin the negative moment controls necessary for lower Moment estimates.
• One-dimensional Log-Concave Comparison and Brunn-Minkowski: Moment comparability for one-dimensional log-concave vectors allows extension from the dyadic lattice to all $|\Theta| \geq 9n/10$3 in $|\Theta| \geq 9n/10$4.

Highlights of the Proof

• For a fixed convex body in isotropic position, the family of symmetric convex sets $|\Theta| \geq 9n/10$5 (defining moment constraints at scale $|\Theta| \geq 9n/10$6) is constructed (Equation (3.1)).
• The measure of their intersection is bounded from below in Gaussian space—a process facilitated by employing the Gaussian correlation inequality (Royen [Royen14]).
• Smaller sublevel sets (where the moment is abnormally small) are shown not to cover all large sections, using volumetric and projection arguments.
• An iterative, greedy but carefully managed selection yields an orthonormal system respecting the constraints.
• The equivalence between $|\Theta| \geq 9n/10$7 norms and Gaussian moment growth is exploited to translate moment estimates to tail bounds (see Vershynin [Vershynin18, Section 2.6]).

Comparison with Previous Work

Past works [Klartag08, GPV11, Paouris05] established subgaussian directions with constants depending polylogarithmically on $|\Theta| \geq 9n/10$8, sometimes with additional symmetry or unconditionality hypotheses. For typical (random) directions, the best available subgaussian constant remains $|\Theta| \geq 9n/10$9 up to lower order corrections [BH15, Milman15]. No prior result provided a dimension-free upper bound valid for almost an entire orthonormal system for arbitrary centered convex bodies of volume one.

The lower moment (supergaussian) estimates follow the spirit of Paouris [Paouris12a], but here they are achieved simultaneously on the same orthonormal family exhibiting subgaussian upper bounds, rather than for potentially different directions.

This result leverages the recent completion of Bourgain's slicing conjecture [KL25, (Bizeul, 12 Jan 2025)], which enables uniform control of isotropic constants for projected measures—a technical cornerstone of the argument.

Numerical Strength and Claims

• The direction set size, $\theta \in \Theta$0, can be replaced by $\theta \in \Theta$1 for any fixed $\theta \in \Theta$2, with constants adjusted accordingly.
• The constants $\theta \in \Theta$3 appearing in the upper/lower inequalities are absolute and independent of the dimension.
• The extension to an entire orthonormal basis is not achieved, and explicit counterexamples are discussed, but it is conjectured that this width (i.e., $\theta \in \Theta$4 directions) could hold for isotropic convex bodies.
• The lower tail bound is sharp in order: for $\theta \in \Theta$5 the lower estimate cannot persist due to extremal cases such as the Euclidean ball.

Theoretical and Practical Implications

This theorem demonstrates that linear functionals along most directions of a high-dimensional convex body behave, in a precise sense, almost like Gaussians, with constants independent of the ambient dimension. The result:

• Advances Asymptotic Convex Geometry: It provides a new dimension-free tool for understanding high-dimensional marginals—critical for phenomena governed by concentration of measure, random projections, and their isoperimetric consequences.
• Increases the Reach of Probabilistic Techniques: The prevalence of subgaussian directions enables wider applicability of machine learning, data analysis, and compressed sensing techniques premised on subgaussianity in random projections and high-dimensional signal processing.
• Suggests Directions for Further Research: The conjecture about extension to an orthonormal basis and the exploration of anti-concentration (small-ball probabilities) remain open, as do possible connections to empirical processes and high-dimensional optimization.

Conclusion

This paper resolves a central open problem in high-dimensional convex geometry, providing explicit, dimension-free Gaussian tail estimates for almost a full orthonormal system of linear functionals on any centered convex body of unit volume. The approach combines geometric, analytic, and probabilistic machinery, incorporating recent advances in the understanding of isotropic constants and negative moment estimates. The resulting prevalence of Gaussian-type behavior in high dimension has broad implications for theoretical and applied mathematics, and invites further investigation into analogous basis results and their applications.

Reference:

Letwin, B., & Mikulincer, D. "Dimension-free Gaussian tail estimates for linear functionals on convex bodies" (2605.10939).