Expressing intrinsic volumes as rotational integrals
A rotational formula of Crofton-type is derived for intrinsic volumes of a compact subset \(X\subset\mathbb{R}^d\) of positive reach. Motivated by applications in local stereology, the authors address the problem of finding functional \(\alpha_{jk}\) defined on the section of \(X\) with a \(j\)-dimensional linear subspace, and whose rotational average is equal to the intrinsic volume \(V_{d-j+k}(X)\) of \(X\), for \(j=0,1,\dots ,d\), and \(k=0,1,\dots ,j\). Solutions are derived for general \(j,k\). A simplified version is obtained in the case \(k=j\), while, for \(k<j\), there is a more explicit formula if \(X\) is the compact closure of an open subset of \(\mathbb{R}^d\), and \(\partial X\) is a special \((d-1)\)-dimensional manifold of class \(C^2\). When \(k=j-1\), the result can be further simplified.
scientific article; zbMATH DE number 5710830