  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.
