Sudakov's Minoration Theorem

• Sudakov's minoration theorem is a foundational result in probability and convex geometry that links the richness of an index set with unavoidable lower bounds on process suprema and covering numbers.
• It extends classical results for Gaussian processes to log-concave and dependent structures, with significant implications in empirical process theory and high-dimensional statistics.
• Techniques such as chaining, tail inequalities, and geometric decompositions underpin its proofs and have advanced understanding in stochastic PDEs and metric entropy problems.

Sudakov’s minoration theorem is a collection of fundamental results in probability theory, convex geometry, and stochastic process theory that provide lower bounds for the supremum of stochastic processes (often Gaussian or log-concave), as well as sharp metric entropy inequalities for convex bodies in high-dimensional spaces. The Sudakov principle links the “richness” or separation of an index set—whether in process theory or convex geometry—to unavoidable lower bounds on the complexity or magnitude of the supremum or covering numbers. Its influence spans several mathematical disciplines, including geometric functional analysis, empirical process theory, and high-dimensional statistics.

1. Formulations of Sudakov's Minoration Theorem

The classical Sudakov minoration theorem asserts that for a centered Gaussian process $\{X_t : t \in T\}$ indexed by a finite set $T$ with metric $d(s, t) = (\mathbb{E}|X_t - X_s|^2)^{1/2}$, the expected supremum is bounded below in terms of the separation and cardinality of $T$: $$\mathbb{E}\sup_{t\in T} X_t \geq c \cdot u \, \sqrt{\log N(T, d, u)}$$ where $N(T, d, u)$ is the $u$-covering number and $c>0$ is universal. If all pairwise distances exceed $u$ and $|T|=n$, then

$$\mathbb{E} \sup_{t\in T} X_t \geq c u \sqrt{\ln n}.$$

This lower bound captures the necessity, imposed by the metric structure, for “large” suprema even in the absence of independence.

Generalizations extend the result from Gaussian increments to log-concave random vectors, canonical processes with regular moments, radial product structures, and processes with certain dependency. In convex geometry, Sudakov’s inequality relates the covering numbers of convex bodies by Euclidean balls to the mean width.

2. Minoration for Gaussian, Log-Concave, and Radial Measures

For Gaussian processes, independence and the properties of the covariance structure provide the canonical metric $d_2(s, t) = (\mathbb{E}|X_s - X_t|^2)^{1/2}$, and the minoration is sharp and universal.

For log-concave vectors, Sudakov-type minoration states that if $X$ is a log-concave random vector in $\mathbb{R}^d$ and $T \subset \mathbb{R}^d$ has $|T| > e^p$ and for all distinct $t, s \in T$, $\|\langle t-s, X \rangle\|_p > A$, then

$$\mathbb{E}\sup_{t \in T}\langle t, X \rangle \geq kA$$

where $k$ may depend on structural properties such as independence, symmetry, or unconditionality of $X$ (Latała, 2013, Bednorz, 2014, Bednorz, 2022). Product and independent coordinate cases admit universal constants, while dependencies (measured via support overlap or VC-dimension) require further combinatorial-geometric techniques such as “common witnesses.”

For canonical processes $X_t = \sum t_i X_i$ where $\{X_i\}$ have regular moment growth, the minoration theorem guarantees (for suitable regularity parameters)

$\mathbb{E}\sup_{s, t \in T}(X_s-X_t) \geq K u$

with $K$ depending only on moment regularity and for $|T| > e^p$ with separation in the $L_p$-metric (Latała et al., 2014).

Product-structured radial log-concave measures—i.e., densities of the form $\exp(-\sum_k U_k(|x_k|^{p_k}))$—also support Sudakov minoration, especially via reductions to independent cases and generic chaining for sharp upper and lower bounds (Bednorz, 2022).

3. Dual Minoration and Applications in Convex Geometry

Sudakov’s inequality in convex geometry provides a fundamental relation between the metric entropy of origin-symmetric convex bodies and their mean width: $$\log N(K, t B_2^n) \leq c n \left(\frac{w(K)}{t}\right)^2,$$ where $N(K, tB_2^n)$ is the minimal number of Euclidean balls needed to cover $K$, and $w(K)$ is the mean width (Naszódi, 2016). The dual minoration form provides an analogous upper bound for covering the Euclidean ball by homothetic copies of $K$: $$\log N(B_2^n, t K) \leq c n \left(\frac{w(K^*)}{t}\right)^2,$$ where $K^*$ is the polar body. This duality connects to deep phenomena in high-dimensional geometry and is pivotal for entropy, covering, and illumination problems.

Recent developments have placed Sudakov’s minoration within the context of magnitude theory: for a convex body $K$ with $N$ well-separated points, the intrinsic volume $V_1(K)$ satisfies $V_1(K) \gtrsim \sqrt{\log N}$, using arguments based on bounding the metric magnitude via Holmes–Thompson volumes (Meckes, 2022). This approach offers new conceptual proofs bypassing classical functional analytic techniques.

4. Techniques and Methods of Proof

Sudakov’s minoration has been established via a range of methods:

• Chaining and Majorizing Measures: Chaining controls the suprema of stochastic processes via hierarchies of $\epsilon$-nets, while Talagrand’s majorizing measure theorem gives matching upper bounds. The minoration constitutes the lower bounding half, implying that the supremum cannot be too small once the index set is sufficiently rich (Liu et al., 30 Jul 2025, Latała et al., 2014, Latała, 2013).
• Tail and Concentration Inequalities: Precise estimates for tail probabilities of log-concave vectors (e.g., via Bobkov–Nazarov inequalities) underpin generalized results beyond the Gaussian case (Bednorz, 2014).
• Geometric and Combinatorial Decomposition: Reduction to thin or disjoint supports, control by VC-dimension of support families, use of witness vectors, and Bernoulli comparisons are key in the presence of dependent or overlapping increments (Bednorz, 2014).
• Dimension and Structure Reduction: For convex bodies, dimension reduction is integral to establishing sharp minoration in the dual Sudakov context, involving sophisticated combinatorial and geometric arguments such as Johnson–Lindenstrauss-style projections or combinatorial dimension for cubes (Mendelson et al., 2016).
• Convex Geometry and Entropy Techniques: For covering arguments, the use of Gaussian measure and maximal separated sets combined with volume/probabilistic estimates allows for entropy bounds and duality principles (Naszódi, 2016).

5. Impact, Extensions, and Applications

Sudakov’s minoration theorem underlies a host of modern results in geometric functional analysis, probability, and information theory:

• Stochastic PDE Analysis: The theorem yields matching lower bounds for the roughness and long-term asymptotic growth in the solution of stochastic partial differential equations (SPDEs), such as the linear stochastic fractional heat equation. In these contexts, Talagrand’s chaining gives upper bounds, Sudakov’s minoration ensures sharpness of the rates for suprema and Hölder coefficients, and both together establish precise asymptotics (Liu et al., 30 Jul 2025).

  Typical quantitative bound in the SFHE context:

  $$\mathbb{E}\bigl[ \sup_{x\in[-L,L]} u(T,x) \bigr] \gtrsim T^{\frac{2H+\alpha-2}{2\alpha}} \sqrt{\ln \left(\frac{L}{T^{1/\alpha}}\right)}$$

  where $u(t, x)$ is the solution of the SFHE, providing critical information for understanding Hölder regularity and intermittency.

• Metric Entropy and Covering Problems: In convex geometry, the theorem answers fundamental questions about the efficiency of translative coverings, with strong implications for Rogers’ covering density problem and illumination conjectures (Naszódi, 2016).
• Duality and Information-Theoretic Extensions: Recent work interprets Sudakov’s minoration in the dual or “soft” sense, connecting supremum bounds to mutual information in optimal transport and detection problems, such as high-dimensional spiked tensor models. Here, soft-max (free energy) lower bounds, derived via convex geometric methods, serve as analogues of classic Sudakov bounds and show phase transition phenomena in statistical signal detection (Liu, 2023).
• Empirical Process Theory and Statistics: The structural control offered by Sudakov’s minoration brings sharp moment comparison, weak/strong moment equivalence, and chaining techniques into probabilistic bounds central for empirical process bounding, high-dimensional statistics, and learning theory (Latała, 2013, Latała et al., 2014, Bednorz, 2022).

6. Open Problems and Research Directions

Several central questions remain:

• Universal Constants for Log-Concave Random Vectors: While the conjecture holds for many special cases (product measures, rotational/invariance, unconditional log-concave vectors modulo logarithmic factors), proving Sudakov’s minoration uniformly for arbitrary log-concave random vectors with universal constants remains open (Latała, 2013, Bednorz, 2014, Bednorz, 2022).
• Dimension Reduction for Arbitrary Convex Bodies: Completing the dimension/separation reduction step universally in the generalized dual Sudakov program is currently unresolved for bodies lacking symmetry or “niceness,” such as $B_1^n$ (Mendelson et al., 2016).
• Connections to Magnitude and Intrinsic Volumes: The emerging bridge between metric invariants, such as magnitude and classical intrinsic volumes, via Sudakov’s principle prompts further paper into new classes of metric entropy inequalities and their extremal cases (Meckes, 2022).
• Further Extensions to Dependent Structures and Nonlinear Processes: Current methods handle one-unconditional and certain dependencies, but broadening the scope to more intricate dependency structures or to non-log-concave tails is under active exploration (Bednorz, 2014, Liu, 2023).
• Sharpness in High-Dimensional Detection and Free Energy: The utility of soft-minoration theorems and their tightness in phase transition thresholds and statistical error exponents open new research in high-dimensional inference (Liu, 2023).

7. Summary Table: Main Sudakov Minoration Results

Setting	Minoration Principle	Key Constant(s) or Factors
Gaussian Process	$\mathbb{E}\sup X_t \geq c u \sqrt{\ln n}$	Universal $c$
Product/Independent Log-Concave	$$\mathbb{E} \sup_{t\in T} \langle t, X\rangle \geq k A$$	Universal $k$
Unconditional Log-Concave	$\mathbb{E} \sup \geq (1/C\log(d+1)) A$	$C$ grows with $\log d$
Radial-Product Log-Concave	$\mathbb{E} \sup \geq (1/K) A$	$K$ universal
Canonical with Regular Moments	$\mathbb{E}\sup (X_s-X_t) \geq K(\alpha) u$	Depends on reg. of moments
Metric Entropy (Convex Geometry)	$\log N(K, t B_2^n) \leq c n (w(K)/t)^2$	$c$ universal
Dual (Polar)	$\log N(B_2^n, tK) \leq c n (w(K^*)/t)^2$	$c$ universal
Magnitude Approach (Intrinsic Volumes)	$V_1(K) \geq C\sqrt{\log N}$	$C$ absolute

Sudakov’s minoration thus constitutes a foundational link between probabilistic separability, convex geometric complexity, and the geometry of high dimensions, with robust formulations across stochastic analysis, geometry, and modern information theory.