A hierarchy of simple valuations in \(\mathbb{R}^3\) (Q5929641)

A frame in Euclidean space \(\mathbb{R}^3\) is a quadruple \((P,e_1,e_2, e_3)\) consisting of a point \(P\) and an orthonormal basis \((e_1,e_2,e_3)\). The author uses suitable functions on frames to define four classes \(M_3\subset M_2\subset M_1\subset M_0\) of simple valuations on the set \({\mathcal D}\) of convex polytopes in \(\mathbb{R}^3\). The definition of the class \(M_i\) involves integration over the \(i\)-dimensional faces. By an earlier result of the author [On multidimensional simple frame valuations, J. Contemp. Math. Anal. 31, No. 4, 75-87 (1997)] the class \(M_0\) represents all simple valuations on \({\mathcal D}\). The author studies the relation between the other classes and the question which simple valuations can be represented within \(M_i\). As an application, he constructs an example of a simple, translation invariant, continuous valuation on \({\mathcal D}\) without a continuous extension to general convex bodies.