Integral identities for the boundary of a convex body (Q6657309)

Let \(K\) is a convex body in \(\mathbf{R}^d\). It was shown in [\textit{R. V. Ambartzumian}, Combinatorial integral geometry. With applications to mathematical stereology. Ed. with an appendix by Adrian Baddeley. Hoboken, NJ: John Wiley \& Sons (1982; Zbl 0492.53043)] that by integration of combinatorial formulae various generalizations of the Pleijel identities can be derived in a rather systematic way (see also [\textit{R. V. Ambartzumian}, Factorization calculus and geometric probability. Cambridge etc.: Cambridge University Press (1990; Zbl 0715.53049)]). This is the following identity: \N\[\N\int_G h(|g\cap K|) \, \mu_{2,1}(dg) = \frac{1}{2} \int_G h'(G\cap K)\, |g\cap K| \cot \alpha_1\, \cot\alpha_2\, \mu_{2,1}(dg), \N\]\Nwhere \(G\) is the set of all lines in the plane, \(g\in G\), \(g\cap K\) is the chord generated by line \(g\) in \(K\), \(h:\) \(\mathbf{R} \to\mathbf{R}\), \(h(0)=0\), and \(\alpha_i\), \(i=1,2\) is the angle between the endpoint \(x_i\) of the chord \(g\cap K\) and the boundary \(\partial K\) of \(K\) (\(\alpha_1\) and \(\alpha_2\) are the angles on the same side of the chord \(g\cap K\)). Note also that \(\mu_{2,1}\) is the measure in the space of lines \(G\) invariant with respect to the rigid motions (translations and rotations). \NThe above identity is satisfied only for planar convex domains whose boundaries do not have line segments. For polygonal domain \(K\) with sides \(a_i\) of length \(|a_i|\) it is necessary to add the term \(\sum_{i=1}^n \int_0^{|a_i|} h(u) du\). Note that formulas (1.2) and (1.3) are obtained from each other by a simple changing of variables. Note that the first two results of the author in this paper (Theorems 1.1 and 1.2) are generalizations of the Ambartsumian-Pleijel identities to an arbitrary dimension. In the second part of the paper the author generalizes well-known formulas of Blachke-Pentkantschin (see [\textit{R. Schneider} and \textit{W. Weil}, Stochastic and integral geometry. Berlin: Springer (2008; Zbl 1175.60003), Theorem 7.2.7]) and [\textit{M. Zähle}, Math. Nachr. 149, 325--340 (1990; Zbl 0725.60014)] with convex bodies with smooth boundaries in arbitrary dimension.