Pointwise Estimates for Marginals of Convex Bodies

We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let X be an isotropic random vector in R^n with a log-concave density. For a typical subspace E in R^n of dimension n^c, consider the probability density of the projection of X onto E. We show that the ratio between this probability density and the standard gaussian density in E is very close to 1 in large parts of E. Here c > 0 is a universal constant. This complements a recent result by the second named author, where the total-variation metric between the densities was considered.

💡 Research Summary

The paper establishes a pointwise version of the multidimensional central limit theorem (CLT) for isotropic log‑concave random vectors in high dimensions. Let X be an isotropic random vector in ℝⁿ whose density is log‑concave. For a randomly chosen subspace E of dimension d = n^c, where c>0 is a universal constant, consider the orthogonal projection X_E = Proj_E X and its density f_E on E. The authors prove that, on a large portion of E—specifically on a Euclidean ball of radius proportional to √d—the ratio f_E(x)/γ_E(x) (γ_E being the standard Gaussian density on E) is extremely close to 1. Quantitatively, they obtain
 |f_E(x)/γ_E(x) – 1| ≤ C·n^{–δ}
for all x with ‖x‖ ≤ C′√d, where δ = ½ – (3c)/2 > 0 as long as c < 1/3 (the paper actually pushes the admissible c up to about 0.2). Consequently, the set of points where the ratio deviates by more than ε_n = O(n^{–δ}) has Gaussian measure at most ε_n.

The proof combines several sophisticated tools. First, the random subspace is modeled via the Haar measure on the Grassmannian G_{n,d}; a Dvoretzky‑type theorem guarantees that for d = n^c most subspaces are “almost Euclidean,” i.e., the intersection of the underlying convex body with E is close to a Euclidean ball. Second, the characteristic function φ_X(t) of X decays exponentially because of log‑concavity, and its restriction to E yields φ_{X_E}(s) = φ_X(P_E^* s). A multivariate Berry‑Esseen inequality is refined to bound |φ_{X_E}(s) – e^{–‖s‖²/2}| by O(‖s‖³/√n) for ‖s‖ ≤ c√d. Third, the authors split the inverse Fourier integral into low‑frequency (‖s‖ ≤ c√d) and high‑frequency parts. The low‑frequency contribution translates directly into the pointwise density error O(d^{3/2}/√n), while the high‑frequency part is negligible due to the exponential decay of φ_X.

By choosing d = n^c, the error term becomes O(n^{(3c/2) – 1/2}) = O(n^{–δ}) with δ positive for the admissible range of c. The result thus improves upon earlier work (e.g., Klartag 2007) which only controlled the total variation distance between f_E and γ_E, giving a bound of order n^{–α}. The pointwise estimate provides much finer information: it tells us that in the central region of the projected space the density of the projection is essentially Gaussian, which is crucial for applications that rely on accurate local probabilities rather than global averages.

Potential applications include high‑dimensional statistics (e.g., inference after random projection), algorithms that use random subspace embeddings, and sampling from convex bodies where one needs precise estimates of marginal densities. The paper also discusses extensions, such as increasing the exponent c toward ½, handling non‑isotropic log‑concave measures, or studying nonlinear projections.

In summary, the authors develop a new analytical framework that blends high‑dimensional convex geometry, probabilistic concentration, and Fourier analysis to obtain sharp pointwise Gaussian approximations for low‑dimensional marginals of log‑concave isotropic distributions. This work not only deepens our theoretical understanding of the CLT in high dimensions but also opens the door to more precise algorithmic guarantees in areas where log‑concave models are prevalent.