Crofton-type Formulae in Rotational Integral Geometry

Abstract: 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 $\mathbb R^n$ from intersections with invariantly integrated $k$-dimensional affine subspaces, where $k=0,\ldots,n-1$ 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 all 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.

Joint work with Emil Dare and Eva B. Vedel Jensen.

mathematical physicsalgebraic geometryalgebraic topologydifferential geometryrepresentation theorysymplectic geometry

Audience: researchers in the topic

Geometry and Physics @ Lisbon