A uniform influence bound for hypergraph properties

For every fixed integer r ≥ 3, we prove that every relabeling-invariant Boolean property of simple r-uniform hypergraphs on n vertices satisfies ${\mathrm{Var}}_p(f)\le C_r{I}_p(f)/(\log n{)}^{r/(r-1)}$. The constant depends only on r, and the bound holds uniformly for all $0<p<1$ without a monotonicity assumption. For increasing properties, it gives the corresponding threshold-width bound with exponent $r/(r-1)$, proving the hypergraph threshold-width conjecture of Friedgut and Kalai.