Intersections and translative integral formulas for boundaries of convex bodies

Let K,L ⊂ IRn be two convex bodies with non-empty interiors and with boundaries ∂K, ∂L, and let χ denote the Euler characteristic as defined in singular homology theory. We prove two translative integral formulas involving boundaries of convex bodies. It is shown that the integrals of the functions t 7→ χ(∂K ∩ (∂L + t)) and t 7→ χ(∂K ∩ (L + t)), t ∈ IRn, with respect to an ndimensional Haar measure of IRn can be expressed in terms of certain mixed volumes of K and L. In the particular case where K and L are outer parallel bodies of convex bodies at distance r > 0, the result will be deduced from a recent (local) translative integral formula for sets with positive reach. The general case follows from this and from the following (global) topological result. Let Kr, Lr denote the outer parallel bodies of K,L at distance r ≥ 0. Establishing a conjecture of Firey (1978), we show that the homotopy type of ∂Kr ∩ ∂Lr and ∂Kr ∩ Lr, respectively, is independent of r ≥ 0 if K◦ ∩ L◦ 6= ∅ and if ∂K and ∂L intersect almost transversally. As an immediate consequence of our translative integral formulas, we obtain a proof for two kinematic formulas which have also been conjectured by Firey.