on GCI for asymmetric sets

Overview

This note studies Lim and Luo’s extension of the Gaussian Correlation Inequality to one nonsymmetric set (Lim & Luo, 2012). The classical GCI requires two centrally symmetric convex sets. Here one set $A$ is allowed to be merely convex with $0\in A$, but the second set must be a centered ball. The tradeoff is sharp in spirit: symmetry is removed from $A$ by imposing radial structure on $B$.

🏷️ From Symmetry to Radial Testing

The classical Gaussian Correlation Inequality says that, for centrally symmetric convex sets $A,B\subset {\mathbb{R}}^d$,

$${\gamma }_d(A\cap B)\ge {\gamma }_d(A){\gamma }_d(B),$$

where ${\gamma }_d$ is standard Gaussian measure. Royen’s proof establishes the full symmetric theorem and even extends it to multivariate gamma distributions (Royen, 2014). The question in Lim and Luo is different: how much symmetry can be removed if the other set is made very regular?

Their first answer is the following radial version.

Lim-Luo Ball Theorem

Let

$$d\mu (x)=\rho (|x|)dx$$

be a radial probability measure on ${\mathbb{R}}^d$ with positive continuous density. If $A\subset {\mathbb{R}}^d$ is convex and $0\in A$, and if $B$ is a Euclidean ball centered at the origin, then

$$\mu (A\cap B)\ge \mu (A)\mu (B).$$

In particular, the result holds for standard Gaussian measure.

The gain is that $A$ need not be symmetric. The loss is that $B$ is no longer an arbitrary symmetric convex body; it is a centered ball. This is the cleanest setting where radial monotonicity can replace symmetry.

🧩 The Functional Lemma

The proof is elementary because it first proves a functional statement. Define the class

$${\mathcal{C}}_d=\left\{f\in C_c({\mathbb{R}}^d,{\mathbb{R}}_{+}):\{f>c\}\text{ is convex for all }c>0,f(x)\le f(0)\right\}.$$

For $f\in {\mathcal{C}}_d$, each radial restriction is decreasing:

$$t\mapsto f(t\theta ),t\ge 0,$$

for every $\theta \in S^{d-1}$. Indeed, if $0,t_2\theta \in \{f>c\}$ and $0<t_1<t_2$, convexity of the superlevel set forces $t_1\theta \in \{f>c\}$. Also, unless $f$ is identically zero,

$$f(0)>\mu (f).$$

Now let $B_t$ be the centered ball of radius $t$ and define

$$\Phi (t)=\mu (f{1}_{B_t})-\mu (f)\mu (B_t).$$

The desired correlation inequality for $f$ and a centered ball is the assertion $\Phi (t)\ge 0$. Differentiating in spherical coordinates gives

$${\Phi }^{\prime }(t)=\rho (t)t^{d-1}{\int }_{S^{d-1}}(f(t\theta )-\mu (f))d\sigma (\theta ).$$

The key point is that the spherical average

$$t\mapsto {\int }_{S^{d-1}}f(t\theta )d\sigma (\theta )$$

is decreasing. Hence ${\Phi }^{\prime }$ can change sign at most once. Since $f(0)>\mu (f)$, $\Phi (t)>0$ for small positive $t$. Since $f$ has compact support,

$$\Phi (t)=\mu (f)(1-\mu (B_t))>0$$

for every sufficiently large finite $t$. The one-crossing property then forces $\Phi (t)\ge 0$ for every $t$.

What Convexity Is Doing

Convexity is not used through a separation theorem or transport map here. It is used to guarantee radial monotonicity of the approximating functions. Once the radial average is monotone, the proof becomes a one-variable sign-change argument.

🌵 Passing From Functions to Sets

To apply the functional lemma to a convex set $A$, first reduce to bounded closed sets by truncation. Then approximate $1_A$ from above by continuous compactly supported functions. Let

$$f_n(x)=1-n\min \left(n^{-1},\mathrm{dist}(x,A)\right).$$

Then $0\le f_n\le 1$, $f_n=1$ on $A$, and $f_n\downarrow 1_A$. Since $0\in A$, we have $f_n(x)\le f_n(0)=1$. The superlevel sets are tubular neighborhoods of $A$:

$$\{f_n>c\}=\left\{x:\mathrm{dist}(x,A)<\frac{1-c}{n}\right\}.$$

These sets are convex because $A$ is convex. Therefore $f_n\in {\mathcal{C}}_d$, and the functional lemma gives

$$\mu (f_n{1}_B)\ge \mu (f_n)\mu (B).$$

Letting $n\to \infty$ and applying monotone convergence yields

$$\mu (A\cap B)\ge \mu (A)\mu (B).$$

This proves the ball theorem. Compared with the optimal-transport proof of Cordero-Erausquin’s Gaussian-type inequalities (Cordero-Erausquin, 2002), the argument is strikingly elementary. Caffarelli’s contraction theorem remains the deeper structural explanation behind several related Gaussian correlation inequalities (Caffarelli, 2000).

🗝️ Ellipsoids Require More Structure

The paper also treats a second nonsymmetric result where $B$ is an axis-aligned ellipsoid, but the measure and the set $A$ are more constrained. Assume

$$\mu =\underset{i=1}{\overset{d}{⨂}}{\mu }_i,d{\mu }_i(x_i)={\rho }_i(|x_i|)d{x}_i,$$

and suppose $A$ has the coordinate-projection property: whenever $x\in A$, every projection of $x$ onto a coordinate hyperplane also lies in $A$. Then for

$$B=\left\{x\in {\mathbb{R}}^d:\frac{x_1^2}{a_1^2}+\cdots +\frac{x_d^2}{a_d^2}\le 1\right\},$$

Lim and Luo prove

$$\mu (A\cap B)\ge \mu (A)\mu (B).$$

The proof is inductive. Slice the ellipsoid by the last coordinate and apply the lower-dimensional statement to each section. The remaining one-dimensional integral is handled by the elementary FKG inequality: two monotone functions on an interval have nonnegative covariance. The coordinate-projection assumption is exactly what ensures the sliced averages have the required monotonicity.

A Nonsymmetric Coordinate-Projection Set

The simplex-like set

$$A=\left\{x\in {\mathbb{R}}^d:x_i\ge 0\text{ for all }i,\sum _{i=1}^d{x}_i\le 1\right\}$$

is convex, contains the origin, and is far from centrally symmetric. Projecting a point of $A$ by setting one coordinate to zero keeps it inside $A$.

⚠️ Why This Is Not the Full GCI

It is tempting to hope that one can remove symmetry from $A$ and still allow every symmetric convex $B$. The note suggests why this is too optimistic. In the ball theorem, the derivative of

$$\Phi (t)=\mu (A\cap B_t)-\mu (A)\mu (B_t)$$

is controlled by spherical averages, and those averages are monotone because the test sets $B_t$ expand radially. A general symmetric convex body does not provide such a one-parameter radial filtration.

For an ellipsoid, one can recover monotonicity only by imposing a product structure on $\mu$ and a coordinate-projection condition on $A$. Without some replacement for radial or coordinate monotonicity, the sign-change proof has no mechanism to prevent oscillation.

📝 Takeaway

The asymmetric result is best read as a controlled exchange:

Removed assumption	Added structure
$A$ need not be centrally symmetric	$B$ is a centered ball
$B$ may be an axis-aligned ellipsoid	$\mu$ is a product measure and $A$ is stable under coordinate projections

Thus the theorem does not supersede the classical GCI. It isolates a regime where the role usually played by symmetry is replaced by monotonicity along radial or coordinate directions.

🔗 See Also

• on Gaussian Correlation Inequality --- Gives the symmetric Gaussian correlation theorem and Royen’s gamma-distribution proof.
• on Sinkhorn’s Theorem --- Provides another example where positivity and monotonicity of a measure-preserving normalization drive a correlation-type conclusion.
• on concentration of information and log-concave distributions --- Log-concavity is the functional background behind the convex superlevel-set arguments used here.

📚 References

🐻  Caffarelli, L.A. 2000. Monotonicity properties of optimal transportation and the FKG and related inequalities. Communications in Mathematical Physics 214(3), 547–563.

🐻  Cordero-Erausquin, D. 2002. Some applications of mass transport to Gaussian-type inequalities. Archive for rational mechanics and analysis 161, 257–269.

🐻  Lim, A.P.C. & Luo, D. 2012. A note on Gaussian correlation inequalities for nonsymmetric sets. Statistics & Probability Letters 82(1), 196–202.

🐻  Royen, T. 2014. A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions.