Kinematic integral formulas for convex bodies

Abstract

In the following we describe some recent developments in integral geometry. The classical integral geometric formulas for convex bodies and the various generalizations of these formulas, for which the reader may consult the books of Hadwiger [1955], [1957] and Santaló [1976], deal with intersecting convex figures. Our aim here is to present results of a different type in two recent branches of integral geometry. In the first case, which was initiated by Hadwiger, one investigates mean value formulas for convex figures which, in contrast to the classical case, have a positive distance. In the other case, which goes back to work of Firey, one considers measures over contact positions of convex figures. Both topics are closely related. As we shall see, the search for integral formulas of the first type that are as general as possible leads one immediately to a natural definition of contact measures of convex bodies. Moreover, since the integral formulas as well as the contact measures involve curvature measures, our considerations also yield results in a third branch of integral geometry, which is concerned with local versions of the classical formulas as they have been obtained by Federer [1959], Schneider [1975], [1978a].