Kinematic and Crofton formulae of integral geometry: recent variants and extensions

The principal kinematic formula and the closely related Crofton formula are central themes of integral geometry in the sense of Blaschke and Santaló. There have been various generalizations, variants, and analogues of these formulae, in part motivated by applications. We give a survey of recent investigations in the spirit of the kinematic and Crofton formulae, concentrating essentially on developments during the last decade. In the early days of integral geometry, the later illustrious geometers S.S. Chern, H. Hadwiger, L.A. Santaló were attracted by Wilhelm Blaschke’s geometric school and all spent some time with him in Hamburg. There, the young Santaló wrote his work (Santaló 1936) on the kinematic measure in space, studying various mean values connected with the interaction of fixed and moving geometric objects and applying them to different questions about geometric probabilities. Fourty years later, when Santaló’s (1976) comprehensive book on integral geometry appeared, the principal kinematic formula, which is now associated with the names of Blaschke, Santaló and Chern, was still a central theme of integral geometry, together with its generalizations and analogues. At about the same time, the old connections of integral geometry with geometric probabilities were deepened through the use that was made of kinematic formulae, Crofton formulae and integral geometric transformations in stochastic geometry, for example in the theoretical foundations of stereology under invariance assumptions. To get an impression of this, the reader is referred to the books of Matheron (1975), Schneider and Weil (1992, 2000). Integral geometry has also begun to play a role in statistical physics, see Mecke (1994, 1998). Motivated by demands from applications, but also for their inherent geometric beauty, kinematic formulae of integral geometry and their ramifications have continuously remained an object of investigation. In the following, we give a survey of recent progress. We concentrate roughly on the period since 1990, since much of the earlier development is covered by the survey articles of Weil (1979) and Schneider and Wieacker (1993). To the bibliographies of these articles and of Schneider and Weil (1992) we refer for the earlier literature.