Kinematic formulas for area measures

We obtain a Principal Kinematic Formula and a Crofton Formula for surface area measures of convex bodies, both involving linear operators on the vector space of signed measures on the unit sphere $S^{d-1}$. These formulas are related to a localization of Hadwiger's Integral Geometric Theorem. The operators, mentioned above, will be shown to be compositions of spherical Fourier transforms originating in the work of Koldobsky. As an application of our Crofton Formula, we will find an extension of Koldobsky's orthogonality relation for such transforms from the case of even spherical functions to centered functions.