Minkowski algebra. I: A convolution theory of closed convex sets and relatively open convex sets (Q1595328)

Let \(\mathbf V\) be a finite dimensional real vector space, and let \(\mathcal C\) and \(\mathcal O\) denote the families of closed and relatively open convex sets in \(\mathbf V\), respectively. As defined here, the Minkowski algebra of convex sets is the vector space \({\mathbf R}({\mathbf V},{\mathcal C}\cup {\mathcal O})\) generated by the indicator functions of sets in \({\mathcal C}\cup {\mathcal O}\), with multiplication induced by Minkowski (vector) addition. However, there is an immediate difficulty with the latter. With two sets of the same kind (both closed, or both relatively open) there is no problem, and so corresponding subalgebras do exist. However, with mixed sums there is no obviously ``correct'' definition; one main object of this paper is to provide the appropriate framework in which such sums are permitted. This is achieved by constructing homomorphisms between the two subalgebras \({\mathbf R}({\mathbf V},{\mathcal C})\) and \({\mathbf R}({\mathbf V},{\mathcal O})\), which are related to the Euler characteristic. The author ends by looking at the Euler-Radon transform induced by affine Grassmannians, solving a syzygy problem posed by \textit{D. A. Klain} an \textit{G.-C. Rota} [Introduction to geometric probability, Cambridge University Press (1997; Zbl 0896.60004)].