Intersections and translative integral formulas for boundaries of convex bodies (Q2729620)

The authors prove two formulae of translative integral geometry for convex bodies \(K\), \(L\) (with nonempty interiors) in \(\mathbb{R}^n\). The first of these says that NEWLINE\[NEWLINE\begin{multlined} \int_{\mathbb{R}^n}\chi(\partial K\cap (\partial L + t)){\mathcal H}^n(dt)= \\ = (1 + (-1)^n)\sum^{n-1}_{k=1}{n\choose k}\{V(K[k],-L[n - k]) + (-1)^{k-1} V(K[k], L[n-k])\},\end{multlined}NEWLINE\]NEWLINE and the second formula is similar, with \(\partial L\) in the integrand replaced by \(L\). Here, \(\chi\) is the Euler characteristic, \({\mathcal H}^n\) denotes \(n\)-dimensional Lebesgue measure, and the function \(V\) of \(n\) convex bodies is the mixed volume. Replacing \(L\) by \(gL\) with \(g\in SO(n)\) and integrating over all \(g\) with respect to the Haar measure on \(SO(n)\), one obtains two kinematic formulae, which were conjectured by Firey in 1978. These kinematic formulae can also be deduced from work of J. H. G. Fu, but not the translative ones. Major difficulties for a proof stem from the fact that \(\partial K\) and \(\partial L\) are in general not of positive reach and that the Euler characteristic of \(\partial K\cap (\partial L +t)\) need not be defined for all \(t\). These difficulties are overcome by considering the outer parallel bodies \(K_r, L_r\) at distance \(r\geq 0\). For \(r > 0\), the boundaries of \(K_r\), and \(L_r\) are of positive reach, hence the formulae can be deduced from a result of \textit{J. Rataj} and \textit{M. Zähle} [Geom. Dedicata 57, 259-283 (1995; Zbl 0844.53050)]. The general case is then settled by establishing the following topological result, which also proves a conjecture of Firey in a stronger form. If \(K^0\cap L^0\neq \emptyset\) and \(\partial K\) and \(\partial L\) intersect almost transversally, then the homotopy types of \(\partial K_r\cap \partial L_r\) and \(\partial K_r\cap L_r\) (which are Lipschitz manifolds in this case), are independent of \(r\geq 0\).