Discrepancy theory, Tverberg's theorem, and regression depth

We prove new bounds for Tverberg's theorem with tolerance. We show that $N = rt+Θ_{d,r}(t^{1/2-1/(2d)})$, where $N$ is the smallest number such that any set of $N$ points in $\mathbb{R}^d$ has a partition into $r$ parts such that the convex hulls of the parts intersect even if we remove any $t$ of t