Valuations on convex sets of oriented hyperplanes
An oriented hyperplane \(H_{u,t} := \{x \in \mathbb{R}^d \mid \langle x,u\rangle = t\}\), with \(u \in S^{d-1}\) (the unit sphere) and \(t \in \mathbb{R}\), is identified with the point \(\gamma(H_{u,t}) := u + te \in \mathbb{R}^{d+1}\), where \(e\) is a unit vector spanning the last component in \(\mathbb{R}^{d+1} = \mathbb{R}^d \times \mathbb{R}\). Such points \(\gamma(H_{u,t})\) coming from the family \({\mathcal H}^d\) of oriented hyperplanes form a cylinder \(Z = S^{d-1} \times \mathbb{R}\); further, one writes \(\nu(H_{u,t}) := u\) for the normal vector to \(H_{u,t}\). Hyperplanes \(H_{u,t}\) and \(H_{v,s}\) are antipodal if \(v = -u\). A subset \(M\) of \(Z\) which does not contain antipodal hyperplanes is convex if the cone with apex \(0\) (the origin) generated by \(M\) is convex (the situation for those \(M\) which do contain antipodal hyperplanes is more complicated); \({\mathcal K}({\mathcal H}^d)\) is the family of compact convex such subsets, and the Hausdorff metric on compact subsets of \(\mathbb{R}^{d+1}\) induces a metric on \({\mathcal K}({\mathcal H}^d)\). Recall that a mapping \(\varphi: {\mathcal S} \to \mathbb{R}\) on an intersectional class \({\mathcal S}\) of subsets of some set (that is, \(S \cap T \in {\mathcal S}\) if \(S,T \in {\mathcal S}\)) is a valuation if \(\varphi(S \cup T) + \varphi(S \cap T) = \varphi(S) + \varphi(T)\) whenever \(S,T,S \cup T \in {\mathcal S}\). One natural valuation on \({\mathcal K}({\mathcal H}^d)\) is the restriction \(\eta\) of the Haar measure. Others are the spherical analogues \(v_j\) of the intrinsic volumes on the image space \(\nu({\mathcal K}({\mathcal H}^d))\). As an analogue of the well-known characterization by Hadwiger of the intrinsic volumes, the authors conjecture that a continuous rigid motion invariant valuation \(\varphi\) on \({\mathcal K}({\mathcal H}^d)\) is of the form \[ \varphi(K) = c_d\eta(K) + \sum_{j=0}^{d-1} \, c_jv_j(\nu(K)) \] for \(K \in {\mathcal K}({\mathcal H}^d)\), with \(c_0,\ldots,c_d\) constants; they establish this conjecture when \(d = 2\). They also draw attention to related conjectures about the \(v_j\) themselves, which will be involved when (at least) \(d \geq 4\).
scientific article; zbMATH DE number 2138097