Discrepancy of a convex set with zero curvature at one point

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 $\gamma \geq 2$. We consider the $L^{p}$ norm of the discrepancy with respect to translations and rotations of a dilated copy of the set $\Omega$.