Integral Geometry in Minkowski Spaces

Abstract

We investigate the possibility of extending classical integral–geometric results, involving lower-dimensional areas, from Euclidean space to Minkowski spaces (finite-dimensional Banach spaces). Of the two natural notions of area in a Minkowski space, due respectively to Busemann and to Holmes and Thompson, the latter turns out to be the more tractable one. For the Holmes–Thompson area, we derive a translative intersection formula and, in the class of hypermetric Minkowski spaces, full analogues of the Crofton formulae for rectifiable sets and for convex bodies. For the Busemannk-area, we give a short proof of the fact that it coincides, fork-rectifiable sets, with thek-dimensional Hausdorff measure induced by the Minkowski metric.