Extended Concepts of Integral Geometry

In this chapter, we derive further integral geometric formulas for convex bodies. They are related to the principal kinematic formula, either directly or indirectly. As in the latter formula, we have a fixed and a moving set, but in the two subsequent sections we do not consider intersections of both; we form sums of convex bodies or projections of convex bodies to subspaces. First we treat rotation means of Minkowski sums, which will later (Section 8.5) be applied to touching probabilities. The global version is an immediate consequence of the principal kinematic formula; the local version will be proved by techniques similar to those in Sections 5.2 and 5.3. From the formulas for rotation means of sums we deduce projection formulas.

In Section 6.3, we admit (infinite) convex cylinders as moving sets. For these, we derive a local kinematic formula, and we also obtain a formula that combines sections with projections.