Discrepancy estimates for multi-dimensional non-smooth convex bodies: a case study

Abstract

We study $L^2$-averaged discrepancies of finite sequences of points in the torus ${\mathbb{T}}^d$ with respect to translated and dilated copies of convex bodies with non-smooth boundary. Under suitable anisotropic assumptions on the decay of the Fourier transform of the body, we prove matching lower and upper bounds for the averaged discrepancy, obtaining the rate $N^{1-\frac{d+1}{d^2+d-1}}$. This yields an intermediate regime between smooth convex bodies and polytopes and recovers the known exponent $2/5$ in dimension $d=2$. The argument relies on harmonic analysis techniques combined with averaging procedures adapted to the anisotropic setting. As an application, we analyze a class of convex bodies exhibiting mixed geometric features, including flat regions, curved parts, and edges.

34 pages