The Colin de Verdière number and graphs of polytopes

The Colin de Verdière number μ(G) of a graph G is the maximum corank of a Colin de Verdière matrix for G (that is, of a Schrödinger operator on G with a single negative eigenvalue). In 2001, Lovász gave a construction that associated to every convex 3-polytope a Colin de Verdière matrix of corank 3 for its 1-skeleton. We generalize the Lovász construction to higher dimensions by interpreting it as minus the Hessian matrix of the volume of the polar dual. As a corollary, μ(G) ≥ d if G is the 1-skeleton of a convex d-polytope. Determination of the signature of the Hessian of the volume is based on the second Minkowski inequality for mixed volumes and on Bol’s condition for equality.