Intersection theoretic inequalities via Lorentzian polynomials

Abstract

We explore the applications of Lorentzian polynomials to the fields of algebraic geometry, analytic geometry and convex geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property, with respect to $m$-positive classes and Schur classes. We also study its convexity variants -- the geometric inequalities for $m$-convex functions on the sphere and convex bodies. Along the exploration, we prove that any finite subset on the closure of the cone generated by $m$-positive classes can be endowed with a polymatroid structure by a canonical numerical-dimension type function, extending our previous result for nef classes; and we prove Alexandrov-Fenchel inequalities for valuations of Schur type. We also establish various analogs of sumset estimates (Pl\"{u}nnecke-Ruzsa inequalities) from additive combinatorics in our contexts.