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The Colin de Verdi\`ere number $\mu(G)$ of a graph $G$ is the maximum corankof a Colin de Verdi\`ere matrix for $G$ (that is, of a Schr\"odinger operatoron $G$ with a single negative eigenvalue). In 2001, Lov\'asz gave aconstruction that associated to every convex 3-polytope a Colin de Verdi\`erematrix of corank 3 for its 1-skeleton.  We generalize the Lov\'asz construction to higher dimensions by interpretingit as minus the Hessian matrix of the volume of the polar dual. As a corollary,$\mu(G) \ge d$ if $G$ is the 1-skeleton of a convex $d$-polytope.  Determination of the signature of the Hessian of the volume is based on thesecond Minkowski inequality for mixed volumes and on Bol's condition forequality.

The Colin de Verdière number and graphs of polytopes