Intersection theoretic inequalities via Lorentzian polynomials

TL;DR

This paper applies Lorentzian polynomials to establish new intersection inequalities and convexity results in algebraic, analytic, and convex geometry, extending classical inequalities and introducing novel structures.

Contribution

It introduces the rKT property for intersection inequalities, extends polymatroid structures to m-positive classes, and proves Alexandrov-Fenchel inequalities for Schur-type valuations.

Findings

Established the rKT intersection inequalities for m-positive classes.

Extended polymatroid structures to cones generated by m-positive classes.

Proved Alexandrov-Fenchel inequalities for valuations of Schur type.

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.