Stability thresholds for big classes

In 1987, the $\alpha$-invariant theorem gave a fundamental criterion for existence of Kahler-Einstein metrics on smooth Fano manifolds. In 2012, Odaka-Sano extended the framework to $\mathbb{Q}$-Fano varieties in terms of K-stability, and in 2017 Fujita related this circle of ideas to the $\delta$-invariant of Fujita-Odaka. We introduce new invariants on the big cone and prove a generalization of the Tian-Odaka-Sano Theorem to all big classes on varieties with klt singularities, and moreover for all volume quantiles $\tau\in[0,1]$. The special degenerate (collapsing) case $\tau=0$ on ample classes recovers Odaka-Sano's theorem. This leads to many new twisted Kahler-Einstein metrics on big classes. Of independent interest, the proof involves a generalization to sub-barycenters of the classical Neumann-Hammer Theorem from convex geometry.