Upper semicontinuous valuations on the space of convex discs
A valuation on the space \({\mathcal K}^2\) of non-empty compact convex sets in the euclidean plane \(\mathbb{E}^2\) is a function \(\mu:{\mathcal K}^2\to \mathbb{R}\) such that \(\mu(K)+ \mu(L)= \mu(K\cup L)+\mu (K\cap L)\) whenever \(K,L\), \(K \cup L\in {\mathcal K}^2\). An old result of \textit{H. Hadwiger} [see Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer Berlin (1957; Zbl 0078.35703)] states that such a valuation which is rigid motion invariant and continuous with respect to the Hausdorff metric on \({\mathcal K}^2\) is a linear combination of the Euler characteristic, perimeter and area of the sets in \({\mathcal K}^2\). Here, the author relaxes continuity to upper semi-continuity, with the result that a suitable curvature integral of the sets in \({\mathcal K}^2\) must be added to Hadwiger's list.
scientific article; zbMATH DE number 1485589