A numerical criterion for lower bounds on K-energy maps

Let X be a smooth, linearly normal, complex projective (nonlinear) variety. Let B be the Bergman metrics corresponding to this embedding. The speaker will discuss the proof of the following Theorem :

The K-energy of X (with respect to the Fubini-Study metric on X) is bounded below on B if and only if it is bounded below on all one parameter degenerations in B.

This completely reduces the check of the energy bound (at least on B) to a purely finite polyhedral - combinatorial problem in algebraic geometry.