The Geometry of Binary Collisions and Generalized Radon Transforms

Abstract

Elastic collisions are characterized by the conservation of momentum and energy. We consider some geometrical aspects of such collisions, when the energy of one particle can be expressed in terms of the moments as \(\). The geometry of elastic collisions is essential for the regularizing property of the gain term in the Boltzmann equation, which was proved by P.-L. Lions. We show how such results can be deduced from a regularity theorem for generalized Radon transforms by Sogge & Stein. This is possible for \(\) and for \(\); we also show that the same technique cannot be used with other choices of \(\).