On Areas and Integral Geometry in Minkowski Spaces

Abstract: In a recent paper with J. A. Wieacker it was shown that certain integral-geometric formulas of Crofton type for lower-dimensional areas carry over from Euclidean to Minkowski spaces, if the Holmes-Thompson notion of Minkowskian area is employed and the Minkowski spaces are of a special type, called hypermetric. Here we show first that there exist Minkowski spaces for which, among all axiomatically defined Minkowskian areas, only the Holmes-Thompson area allows such generalizations. Then we extend further results from Euclidean translative integral geometry to Minkowski spaces.

Classification (MSC91): 52A21, 46B20, 52A22, 53C65

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