Intersections and translative integral formulas for boundaries of convex bodies

Let K, L C R-n be two convex bodies with non-empty interiors and with bound aries partial derivativeK, at, and let X denote the Euler characteristic as defined in singular homology theory. We prove two translative integral for mulas involving boundaries of convex bodies. It is shown that the integrals of the functions t --> chi(partial derivativeK boolean AND (partial deriva tiveL + t)) and t --> chi(partial derivativeK boolean AND (L + t)), t epsil on R-n, with respect to an lzdimensional Haar measure of R-n can be express ed in terms of certain mixed volumes of K and L. In the particular case whe re K and L are outer parallel bodies of convex bodies at distance r > 0, th e 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 K-r, L-r denote the outer pa rallel bodies of K, L at distance r greater than or equal to 0. Establishin g a conjecture of FIREY (1978), we show that the homotopy type of partial d erivativeK(r) boolean AND partial derivativeL(r) and partial derivativeK(r) boolean AND L-r, respectively, is independent of r greater than or equal t o 0 if K-o boolean AND L-o not equal theta and if partial derivativeK and p artial derivativeL intersect almost transversally. As an immediate conseque nce of our translative integral formulas, we obtain a proof for two kinemat ic formulas which have also been conjectured by FIREY.