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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 Verdi\`ere number and graphs of polytopes