A characterization of affine surface area
A famous theorem by H. Hadwiger says that a real valued functional \(\mu\) on the space \({\mathcal K}^d\) of convex bodies in \(\mathbb{E}^d\) equipped with the Hausdorff topology is the linear combination \[ \mu= \sum^d_{ l=0} c_lW_l\;(c_l\in \mathbb{R},\;l=0,\dots,d) \] of the quermassintegrals \(W_l\) provided that \(\mu\) is assumed to be a valuation (i.e. there is \(\mu(K_1 \cup K_2)+ \mu(K_1\cap K_2)= \mu(K_1)+ \mu(K_2)\) for all \(K_1,K_2\in {\mathcal K}^d\) with \(K_1\cup K_2\in {\mathcal K}^d)\), motion invariant and continuous. T. Ódor posed the problem whether there exists an equiaffine analogon of Hadwiger's theorem in the following form: An equiaffine invariant and upper (resp. lower) semicontinuous valuation \(\mu:{\mathcal K}^d \to\mathbb{R}\) must be of the form \[ \mu= c_0+ c_1V+c_2\Omega\;\bigl(c_0,c_1, c_2\in\mathbb{R},\;c_1\geq 0\text{ (resp. }\leq 0) \bigr) \] where \(V\) is the volume and \(\Omega\) is the (generalized) affine surface area, defined as the Lebesgue integral of the \((d+1)\)-st root of the (a.e. existing) Gaussian curvature of the boundary of the convex body with respect to its \((d-1)\)-dimensional Hausdorff measure. The authors present a positive answer to this problem after the first author has done so in the two-dimensional case (to appear in Abh. Math. Semin. Univ. Hamburg). Their very sophisticated proof follows the basic ideas in this special case but also fully uses the boundary structure of an arbitrary convex body as known by A. D. Aleksandrov. Additionally, an induction argument and approximation by so-called \(\varepsilon\)-smooth bodies should be mentioned in this context. [Remark: As a consequence of the authors' result the equiaffine invariant valuations \(\mu_l\) with \(\mu_l(K):= \int_{\text{bd} K}H_l d\Omega\) \((H_l\) \(l\)-th affine mean curvature of \(\text{bd} K\), \(l=1,\dots, d-1)\) of sufficiently smooth convex bodies \(K\) cannot be extended to all convex bodies in a semicontinuous manner].
scientific article; zbMATH DE number 1389129
scientific article