On areas and integral geometry in Minkowski spaces. (English) Zbl 0889.52005

Schneider, Rolf

Beitr. Algebra Geom. 38, No. 1, 73-86 (1997).

The author continues the investigations of R. Schneider and J. A. Wieacker [Adv. Math. 129, 222-260 (1997)] on integral geometric results in Minkowski spaces and shows the existence of Minkowski spaces for which a Crofton formula for lower-dimensional areas with an associated Crofton measure is only possible (among all axiomatically defined Minkowskian areas) for the Holmes-Thompson area. Other results concern Minkowskian counterparts to translative integral formulae (in Euclidean dean space). They are formulated either in hypermetric spaces or use mixed volumes of associated zonoids.

Reviewer: W.Weil (Karlsruhe)

MSC:

52A21	Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
52A22	Random convex sets and integral geometry (aspects of convex geometry)
46B20	Geometry and structure of normed linear spaces

Keywords:

Minkowski space; Holmes-Thompson area; Crofton formula; hypermetric; zonoid; translative integral geometry