On the Extremals of the Khovanskii-Teissier Inequality

The Khovanskii-Teissier inequality provides the fundamental log-concavity property of intersection numbers of divisors of algebraic varieties, extending the Alexandrov-Fenchel inequality of convex geometry. In this talk I will explain, and attempt to give a largely self-contained proof of, the equality cases of this inequality in the simplest nontrivial case of big nef classes on smooth projective varieties. This was done by Shenfeld and myself in the convex (= toric) setting, and extended by Hu and Xiao to the algebraic setting. I will also aim to explain some obstacles to understanding the full picture of the equality cases in more general situations (beyond the convex setting, where we were able to completely characterize the equality cases).