Integral geometry of flag area measures

Kinematic formulas are one of the main object of study in integral geometry. They express the average of a geometric functional over a group acting on the space of convex bodies, in terms of some other geometric functionals. In the classical kinematic formulas, the intrinsic volumes are considered and the integral can be expressed in terms of all intrinsic volumes only.

In this talk, I shall present a joint work with Andreas Bernig, where we obtain additive kinematic formulas for smooth flag area measures. A flag area measure on a Euclidean vector space is a continuous and translation-invariant valuation (additive functional from the space of convex bodies) with values in the space of signed measures on a fixed flag manifold.

After stating the existence of such additive kinematic formulas, I will consider the particular case of the flag manifold consisting of a unit vector and a linear subspace of fixed dimension which contains the unit vector. We will first give a basis of these flag area measures and interpret geometrically its elements. The kinematic formulas will be obtained after moving to the dual space of flag area measure and studying its structure of algebra.