Blowups with log canonical singularities

We show that the minimum weight of a weighted blow-up of ${\mathbf{A}}^d$ with $\epsilon$-log canonical singularities is bounded by a constant depending only on $\epsilon$ and $d$. This was conjectured by Birkar. Using the recent classification of $4$-dimensional empty simplices by Iglesias-Vali~no and Santos, we work out an explicit bound for blowups of ${\mathbf{A}}^4$ with terminal singularities: the smallest weight is always at most $32$, and at most $6$ in all but finitely many cases.