The purpose of this talk is to give an introduction to rotational integral geometry
and exemplify a number of its core results and their applications. Integral geometry,
introduced by Blaschke in the 1930s, is the theory of invariant measures on
geometric spaces (often Grassmannians) and its application to determine geometric
probabilities. We will start by recalling the kinematic Crofton formula, which allows
us to retrieve certain geometric characteristics (such as volume, surface area and
other intrinsic volumes) of a compact convex set K in ]Rn from intersections with
invariantly integrated k-dimensional affine subspaces, where k = 0, ... , n - 1 is
fixed . Motivated by applications from biology, we suggest a number of variants of
Crofton 's formula, where the intersecting affine spaces are constrained to contain
the origin (and hence are just linear subspaces) or even are all required to contain
a fixed lower-dimensional axis. Corresponding rotational Crofton formulae will be
established and explained . We also show that the set of these formulae is complete
in that they retrieve all possible intrinsic volumes of K . Proofs rely on old and new
Blaschke-Petkantschin theorems, which we also will outline.
This is joint work with Emil Dare and Eva B. Vedel Jensen (Arhus University).