  Inspired by the theory of hyperbolic polynomials and Hodge theory, we develop
the theory of Lorentzian polynomials on cones. This notion captures the
Hodge-Riemann relations of degree zero and one. Motivated by fundamental
properties of volume polynomials of Chow rings of simplicial fans, we define a
class of multivariate polynomials which we call hereditary polynomials. We give
a complete and easily checkable characterization of hereditary Lorentzian
polynomials. This characterization is used to give elementary and simple proofs
of the Heron-Rota-Welsh conjecture for the characteristic polynomial of a
matroid, and the Alexandrov-Fenchel inequalities for convex bodies.
  We then characterize Chow rings of simplicial fans which satisfy the
Hodge-Riemann relations of degree zero and one, and we prove that this property
only depends on the support of the fan.
  Several different characterizations of Lorentzian polynomials on cones are
provided.
