Minkowski Algebra I: a Convolution Theory of Closed Convex Sets and Relatively Open Convex Sets∗

This is the first one of a series of papers on Minkowski algebra. One of purposes of this paper is to set up a general framework so that the mixed volume theory and integral geometry can be developed algebraically in subsequent papers. The so called Minkowski algebra of convex sets is the vector space generated by indicator functions of closed convex sets and relatively open convex sets, where the multiplication is induced by the Minkowski sum of convex bodies. We shall study the homomorphisms induced by the linear maps from the ground vector spaces; the subalgebra spanned by the indicate functions of closed convex sets; the subalgebra spanned by the indicator functions of relatively open convex sets; and the embedding of these subalgebras in the commutative algebra of multivalued functions with multiplicities. With this embedding, we are able to define the Minkowski product for indicator functions of a closed convex set and a relatively open convex set, which could be technically difficult without the embedding; and to classify the multiplicative units of the algebra. Moreover, the group of units with Euler characteristic equal to +1 can be viewed as a vector space isomorphic to the vector space spanned by certain continuous functions, including support functions of convex bodies. Finally, we introduce the Euler-Radon transform for convex chains with respect to the Euler measure, and show that it is injective. This injectivity solves the syzygy problem of [10] on affine Grassmannians.