The extremals of Minkowski's quadratic inequality (Q2119905)

The authors solve a central problem of convex geometry, which is more than a century old: the equality cases in Minkowski's quadratic inequality for mixed volumes. This inequality states that \[ \mathsf{V}(K,L,M,\dots,M)^2\ge \mathsf{V}(K,K,M,\dots,M)\mathsf{V}(L,L,M,\dots,M), \tag{1} \] where \(\mathsf{V}\) denotes the mixed volume and \(K,L,M\) are convex bodies in \(\mathbb{R}^n\). Already Minkowski observed that equality in (1) holds not only in the trivial cases. The inequality (1) is a special case of the Aleksandrov-Fenchel inequalities, and for these (and full-dimensional convex bodies) a conjecture about the equality cases was published by the reviewer in 1985 [\textit{R. Schneider}, in: Discrete geometry and convexity, Proc. Conf., New York 1982, Ann. N.Y. Acad. Sci. 440, 132--141 (1985; Zbl 0567.52004)]. For (1), this is now settled, as follows. A vector \(u\in\mathbb{R}^n\setminus\{0\}\) is called a \(1\)-extreme normal vector of a convex body \(M\) if there do not exist linearly independent normal vectors \(u_1,u_2,u_3\) at the same boundary point of \(M\) such that \(u=u_1+u_2+u_3\). Let \(K,L,M\subset\mathbb{R}^n\) be convex bodies such that \(\mathsf{V}(L,L,M,\dots,M)>0\) (the opposite case is easy to deal with). If \(\dim M=n\), then equality holds in (1) if and only if \(K\) and a suitable (possibly degenerate) homothet of \(L\) have the same supporting hyperplanes in all \(1\)-extreme normal directions of \(M\). If \(\dim M<n\), so that \(M-M\subset w^\perp\) for some unit vector \(w\), then equality holds in (1) if and only if for a suitable dilate \(\tilde L\) of \(L\) it holds that \(K+F(\tilde L,w)\) and \(\tilde L+F(K,w)\) have the same supporting hyperplanes in all \(1\)-extreme normal directions of \(M\) (here \(F(\cdot,w)\) is the support set with outer nomal vector \(w\)). The elaborate proof uses ideas related to hyperbolic quadratic forms and to Dirichlet forms associated with highly degenerate elliptic operators, via the following result. On the Hilbert space \(L^2(S_{B,M,\dots,M})\) (where \(S_{K_1,\dots,K_{n-1}}\), a Borel measure on the unit sphere, is the mixed area measure of \(K_1,\dots,K_{n-1}\) and \(B\) is the unit ball), there is a self-adjoint operator \(\mathcal{A}\) such that the closed quadratic form defined by \(\mathcal{E}(f,g)=\langle f,\mathcal{A}g\rangle\) satisfies \(h_K,h_L\in\operatorname{Dom} \mathcal{E}\) and \(\mathsf{V}(K,L,M,\dots,M)=\mathcal{E}(h_K,h_L)\) (where \(h\) is the support function). The kernel of this operator plays an essential role. Under the assumptions that \(\dim M=n\) and \(\mathsf{V}(L,L,M,\dots,M)>0\), the proof of the main result proceeds in two steps. If equality holds in (1), then there exist \(a\ge 0\) and \(v\in\mathbb{R}^n\) such that \(h_K(x)-ah_L(x)=\langle v,x\rangle\) for all \(x\in \operatorname{supp} S_{M,\dots,M}\). If equality holds in (1) and \(h_K(x)=h_L(x)\) for all \(x\in \operatorname{supp} S_{M,\dots,M}\), then \(h_K(x)=h_L(x)\) for all \(x\in\operatorname{supp} S_{B,M,\dots,M}\). Since the operator \(\mathcal{A}\) cannot be approached directly (if \(M\) is full-dimensional), the authors have to prove additional stability and rigidity results.