$K-$Lorentzian Polynomials, Semipositive Cones, and Cone-Stable EVI Systems

TL;DR

This paper extends Lorentzian polynomial theory to convex cones, establishing new inequalities, geometric structures, and stability criteria for cone-constrained systems with applications in variational analysis and probability.

Contribution

It introduces $K$-Lorentzian polynomials over convex cones, develops cone-restricted inequalities, and links these to stability and negative dependence in cone-constrained dynamics.

Findings

$K$-Lorentzian forms define proper cones with maximal Lorentzian properties.

Cone-restricted Rayleigh inequalities relate to bilinear form acuteness.

$K$-Lorentzian quadratic forms imply cone stability and copositivity.

Abstract

Lorentzian and completely log-concave polynomials have recently emerged as a unifying framework for negative dependence, log-concavity, and convexity in combinatorics and probability. We extend this theory to variational analysis and cone-constrained dynamics by studying $K$-Lorentzian and $K$-completely log-concave polynomials over a proper convex cone $K\subset\mathbb{R}^n$. For a $K$-Lorentzian form $f$ and $v\in\operatorname{int}K$, we define an open cone $K^\circ(f,v)$ and a closed cone $K(f,v)$ via directional derivatives along $v$, recovering the usual hyperbolicity cone when $f$ is hyperbolic. We prove that $K^\circ(f,v)$ is a proper cone and equals $\operatorname{int}K(f,v)$. If $f$ is $K(f,v)$-Lorentzian, then $K(f,v)$ is convex and maximal among convex cones on which $f$ is Lorentzian. Using the Rayleigh matrix $M_f(x)=\nabla f(x)\nabla f(x)^T - f(x)\nabla^2 f(x)$, we obtain cone-restricted Rayleigh inequalities and show that two-direction Rayleigh inequalities on $K$ are equivalent to an acuteness condition for the bilinear form $v^T M_f(x) w$. This yields a cone-restricted negative-dependence interpretation linking the curvature of $\log f$ to covariance properties of associated Gibbs measures. For determinantal generating polynomials, we identify the intersection of the hyperbolicity cone with the nonnegative orthant as the classical semipositive cone, and we extend this construction to general proper cones via $K$-semipositive cones. Finally, for linear evolution variational inequality (LEVI) systems, we show that if $q(x)=x^T A x$ is (strictly) $K$-Lorentzian, then $A$ is (strictly) $K$-copositive and yields Lyapunov (semi-)stability on $K$, giving new Lyapunov criteria for cone-constrained dynamics.