An integral geometric theorem for simple valuations (Q1403516)

Let \({\mathcal K}^n\) be the space of convex bodies (understood as nonempty compact convex subsets of \(\mathbb R^n\)), endowed with the Hausdorff metric; let \(\mu\) be the (suitably normalized) Haar measure on the group \(G_n\) of rigid motions of \(\mathbb R^n\), and \(\mu_q\) the (suitably normalized) motion invariant measure on the space \({\mathcal E}^n_q\) of \(q\)-dimensional affine subspaces of \(\mathbb R^n\). A map \(\phi:{\mathcal K}^n \rightarrow \mathbb{R}\) is a valuation if \[ \phi(K \cup M) + \phi(K \cap M) = \phi(K) + \phi(M) \] for any \(K,M \in {\mathcal K}^n\) with \(K \cup M \in {\mathcal K}^n\). If, moreover, \(\phi(K) = 0\) whenever \(\dim K < n\), then \(\phi\) is called a simple valuation. The following theorem is due to Hadwiger: Theorem 1. Let \(\phi: {\mathcal K}^n \rightarrow \mathbb{R}\) be a continuous valuation. Then \[ \int_{G_n} \phi(K \cap gM) d \mu(g)= \sum_{k=0}^n \phi_k(K)V_{n-k}(M) \] for \(K,M \in {\mathcal K}^n\), where \[ \phi_k(K)=\int_{{\mathcal E}^n_{n-k}} \phi(K \cap E) \,d\mu_{n-k}(E). \] The author proves Theorem 2 which is an analogue of Theorem 1. It concerns continuous simple valuations and the group of translations instead of \(G_n\). Roughly speaking, Theorem 2 is the following: Theorem 2. Let \(\phi: {\mathcal K}^n \rightarrow \mathbb{R}\) be a continuous simple valuation. Then \[ \int_{\mathbb R^n}\phi(K \cap (M+x)) d\lambda(x) = \phi(K)V_n(M) + \int_{S^{n-1}}f_{K,\phi}(u) S_{n-1}(M,du), \] where \(f_{K,\phi}\) is some odd function on \(S^{n-1}\).