On weighted parallel volumes
Let \(K\) be a non-empty compact convex subset of \(\mathbb{R}^d\) (a convex body). Let \(B^d\) denote the \(d\)-dimensional unit ball and \(V_d\) the Lebesgue measure. The volume \(V_d(K+ rB^d)\) of the parallel body associated with \(K\) and \(r\in\mathbb{R}_{\geq 0}\) is a polynomial function in \(r\) of degree \(\leq d\), \[ V_d(K+ rB^d)= \sum^d_{j=0} a_{d-j} r^{d-j}. \] Let \(\kappa_j\) denote the volume of \(B^j\). The identity \(a_{d-j}= \kappa_{d-j} V_j(K)\) defines the `intrinsic volumes' \(V_0(K),\dots, V_d(K)\) of \(K\). So \[ W(K):= \sum^d_{j=0} V_j(K) \] assigns a real number to each convex body. The mapping \(W\) is called the Wills functional. H. Hadwiger proved \[ W(K)= \int_{\mathbb{R}^d} e^{-\pi d(K,x)^2}\,dx,\tag{H} \] where \(d(K, x)\) denotes the Euclidean distance of \(x\in\mathbb{R}^d\) to \(K\). Further, \(W(K)\) is the expectation value of an appropriate random variable. The right-hand expression in (H) makes sense for arbitrary bodies. The author studies generalizations of the Wills functional when \(B_j\) is replaced by a convex body and \(V_d\) is a `signed measure'. In particular the author investigates analogues to Hadwiger's identity (H) and gives sufficient conditions on the signed measure such that the resulting `weighted parallel volumes' are invariant under translations, continuous, additive.
scientific article; zbMATH DE number 5601581