# Discrepancy of a convex set with zero curvature at one point

## TL;DR

This paper investigates the discrepancy behavior of convex sets with a specific zero curvature point, focusing on how their geometric properties influence the $L^p$ discrepancy under transformations.

## Contribution

It provides a detailed analysis of the discrepancy for convex bodies with a zero curvature point, especially near the boundary described by a power law, extending understanding of geometric discrepancy.

## Key findings

- Discrepancy depends on the curvature properties at the boundary.
- Zero curvature at a point affects the discrepancy rate.
- Results apply to convex bodies with boundary locally modeled by $|x|^{eta}$.

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