Abstract
The investigation of functions on convex bodies which are valuations, or additive in Hadwiger’s sense, has always been of interest in particular parts of geometric convexity, and it has seen some progress in recent years. The occurrence of valuations in the theory of convex bodies can be traced back to the notion of volume in two essentially different ways. Firstly, the volume of convex bodies, being the restriction of a measure, is itself a valuation. This valuation property carries over to the functions which are deduced from volume in the Brunn-Minkowski theory, namely to mixed volumes, quermassintegrals, surface area functions, and others. Hadwiger’s celebrated characterizations of the quermassintegrals by the valuation and other properties were the culmination of a series of papers on valuations and at the same time the starting point for various subsequent investigations of functionals with similar properties.
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