Discrepancy of a convex set with zero curvature at one point

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a convex body with everywhere positive curvature except at the origin and with the boundary $\mathrm{\partial }\mathrm{\Omega }$ as the graph of the function $y=\mathrm{\mid }x{\mathrm{\mid }}^{\gamma }$ 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 $\mathrm{\Omega }$.