Discrepancy of a convex set with zero curvature at one point

Abstract

Let $\Omega \subset {\mathbb{R}}^d$ be a convex body with everywhere positive curvature except at the origin and with the boundary $\partial \Omega$ as the graph of the function $y=|x{|}^{\gamma }$ in a neighborhood of the origin with $\gamma \ge 2$. We consider the $L^p$ norm of the discrepancy with respect to translations and rotations of a dilated copy of the set $\Omega$.