Log-Semiconvex Functions: Theory & Applications

• Log-semiconvex functions are smooth, positive functions on ℝⁿ whose logarithms have a quantified lower curvature bound, bridging log-convexity and non-convexity.
• They underpin deviation inequalities under Gaussian measures, with key tools like the Ornstein–Uhlenbeck semigroup and Ehrhard's inequality enhancing tail behavior analysis.
• Analytical results here yield sharp tail bounds and improved regularization properties, connecting to Talagrand’s conjecture and advancing probabilistic and geometric methods.

A log-semiconvex function is a smooth, positive-valued function on $\mathbb{R}^n$ whose logarithm exhibits a quantified lower bound on its curvature, controlled by a parameter $\beta \ge 0$. These functions interpolate between log-convexity ($\beta = 0$) and non-convex behaviors, providing a key analytic class for deviation inequalities under Gaussian measures, with deep connections to semigroup regularization, the Ornstein–Uhlenbeck (OU) flow, and the Gaussian analogues of Talagrand’s conjecture (Gozlan et al., 2017).

1. Definition and Characterization

Let $g:\mathbb{R}^n\to(0,\infty)$ be smooth. The function $g$ is log-semiconvex with constant $\beta \ge 0$ (or $\beta$-log-semiconvex) if

$$\mathrm{Hess}(\log g(x)) \succeq -\beta\,\mathrm{Id}, \quad \text{for all } x\in\mathbb{R}^n,$$

where $\mathrm{Hess}$ is the Hessian and $\succeq$ denotes matrix inequality in the Loewner order. Equivalently, writing $\beta \ge 0$0 for some $\beta \ge 0$1, this is $\beta \ge 0$2. The case $\beta \ge 0$3 recovers log-convexity, i.e., convexity of $\beta \ge 0$4.

2. Canonical Examples

Three principal classes under the standard Gaussian measure $\beta \ge 0$5 are documented:

Function	$\beta \ge 0$6 value	Notable features
$\beta \ge 0$7	$\beta \ge 0$8	Log-convex (linear exponent, $\beta \ge 0$9)
$\beta = 0$0, Ornstein–Uhlenbeck smoothing	$\beta = 0$1 for $\beta = 0$2	Smoothing of $\beta = 0$3 preserves log-semiconvexity
$\beta = 0$4	$\beta = 0$5	“Tilted Gaussians”; Hessian $\beta = 0$6

Ornstein–Uhlenbeck smoothing: For $\beta = 0$7 nonnegative, the OU semigroup $\beta = 0$8 satisfies

$\beta = 0$9

and is thus $g:\mathbb{R}^n\to(0,\infty)$0–log–semiconvex with $g:\mathbb{R}^n\to(0,\infty)$1.

3. Deviation Inequalities

A fundamental property of log-semiconvex functions is the existence of sharp deviation inequalities under $g:\mathbb{R}^n\to(0,\infty)$2. For $g:\mathbb{R}^n\to(0,\infty)$3, $g:\mathbb{R}^n\to(0,\infty)$4, and $g:\mathbb{R}^n\to(0,\infty)$5,

$g:\mathbb{R}^n\to(0,\infty)$6

where $g:\mathbb{R}^n\to(0,\infty)$7 for universal $g:\mathbb{R}^n\to(0,\infty)$8. Setting $g:\mathbb{R}^n\to(0,\infty)$9, $g$0,

$g$1

For $g$2 (log-convexity), a sharp form is available: $g$3 where $g$4 is the upper tail of the standard normal. The standard Gaussian estimate $g$5 implies the optimality in exponential–root behavior: $g$6

4. Proof Approaches and Analytical Tools

Two distinctive arguments provide the convex-case (zero semiconvexity) deviation bounds:

A) Ehrhard-Concavity and Fenchel–Moreau Duality:

Define sublevel sets $g$7 with $g$8. Convexity of $g$9 and Ehrhard’s inequality yield concavity and monotonicity of $\beta \ge 0$0. Analysis via integration by parts and Fenchel–Legendre conjugation shows

$\beta \ge 0$1

establishing the sharp tail via the normal CDF.

B) Monotone Rearrangement and One-Dimensional Bound:

Transport $\beta \ge 0$2 via $\beta \ge 0$3 to obtain push-forward $\beta \ge 0$4 with distribution function $\beta \ge 0$5. Analysis of $\beta \ge 0$6 using convexity/concavity properties gives

$\beta \ge 0$7

so that $\beta \ge 0$8.

A general “semi-convex comparison” lemma states: for $\beta \ge 0$9 with $\beta$0 and $\beta$1,

$\beta$2

This lemma underpins the full $\beta$3–log–semiconvex deviation estimates.

5. Connection to Talagrand’s Conjecture and Ornstein–Uhlenbeck Regularization

The classical Talagrand conjecture in discrete settings concerns the $\beta$4 smoothing properties of biased convolution on the Boolean hypercube. Its continuous analogue involves regularization by the Ornstein–Uhlenbeck semigroup $\beta$5. While hypercontractivity is trivial at $\beta$6, it is shown that

$\beta$7

Combined with general tail bounds for $\beta$8–log–semiconvex functions, this yields

$\beta$9

proving that $$\mathrm{Hess}(\log g(x)) \succeq -\beta\,\mathrm{Id}, \quad \text{for all } x\in\mathbb{R}^n,$$0 improves tail behavior even starting from $$\mathrm{Hess}(\log g(x)) \succeq -\beta\,\mathrm{Id}, \quad \text{for all } x\in\mathbb{R}^n,$$1. Lehec (Gozlan et al., 2017) established the sharp asymptotic $$\mathrm{Hess}(\log g(x)) \succeq -\beta\,\mathrm{Id}, \quad \text{for all } x\in\mathbb{R}^n,$$2 decay with $$\mathrm{Hess}(\log g(x)) \succeq -\beta\,\mathrm{Id}, \quad \text{for all } x\in\mathbb{R}^n,$$3-independent constants.

6. Sharpness and Further Corollaries

The inequality $$\mathrm{Hess}(\log g(x)) \succeq -\beta\,\mathrm{Id}, \quad \text{for all } x\in\mathbb{R}^n,$$4 is exactly sharp, being achieved for linear functions $$\mathrm{Hess}(\log g(x)) \succeq -\beta\,\mathrm{Id}, \quad \text{for all } x\in\mathbb{R}^n,$$5. Preservation of log-convexity under $$\mathrm{Hess}(\log g(x)) \succeq -\beta\,\mathrm{Id}, \quad \text{for all } x\in\mathbb{R}^n,$$6 yields, for $$\mathrm{Hess}(\log g(x)) \succeq -\beta\,\mathrm{Id}, \quad \text{for all } x\in\mathbb{R}^n,$$7 log-convex: $$\mathrm{Hess}(\log g(x)) \succeq -\beta\,\mathrm{Id}, \quad \text{for all } x\in\mathbb{R}^n,$$8 independently of $$\mathrm{Hess}(\log g(x)) \succeq -\beta\,\mathrm{Id}, \quad \text{for all } x\in\mathbb{R}^n,$$9. Furthermore, by applying monotone push-forward maps (e.g., cumulative distribution transforms to the exponential or chi-square), results extend to additional structured measures. These conclusions strictly enhance previous deviation inequalities and are summarized in Corollary 1.2 and Corollary 1.4 (Gozlan et al., 2017).

7. Geometric and Probabilistic Implications

Ehrhard’s inequality (Gaussian Brunn–Minkowski) is pivotal, not only ensuring concavity of the quantile transform $\mathrm{Hess}$0 but also classical Gaussian facts such as median $\mathrm{Hess}$1 mean for convex $\mathrm{Hess}$2. Attempts to extend purely geometric rearrangement methods from the convex ($\mathrm{Hess}$3) case to the general log-semiconvex case ($\mathrm{Hess}$4) encounter obstacles; local minima in $\mathrm{Hess}$5 can disrupt required semiconvexity of the induced monotone transport $\mathrm{Hess}$6, as elucidated by van Handel. Consequently, sharp tail bounds for $\mathrm{Hess}$7–log–semiconvex functions for $\mathrm{Hess}$8 necessarily depend on semigroup or stochastic localization techniques rather than geometric rearrangement (Gozlan et al., 2017).