2. Normal approximation over convex sets🔗

Throughout this section E=\mathbb R^d has its Euclidean norm. For A\subseteq E, write d_A(x)=\inf_{a\in A}\|x-a\|,\qquad A^\varepsilon=\{x:d_A(x)<\varepsilon\}.

The proof has four layers. Convex distance produces smooth indicator approximations; signed distance and coarea reduce Gaussian shells to boundary measure; Ball's projection argument bounds that boundary measure by the sharp order d^{1/4}; and a Gaussian replacement induction, followed by covariance whitening, proves the multivariate limit theorem.

1. 2.1. Convex distance and smooth cutoffs
2. 2.2. Gaussian companions and rotation
3. 2.3. Whitening and covariance normalization
4. 2.4. Signed distance and coarea
5. 2.5. Ball's Gaussian perimeter theorem
6. 2.6. Smoothing convex indicators
7. 2.7. Gaussian density calculus
8. 2.8. Taylor remainders and angle splitting
9. 2.9. The identity-covariance replacement induction
10. 2.10. Bentkus's convex-set theorem