Tensor Valuations and Their Local Versions

Abstract

The intrinsic volumes, recalled in the previous chapter, provide an array of size measurements for a convex body, one for each integer degree of homogeneity from 0 to n. For measurements and descriptions of other aspects, such as position, moments of the volume and of other size functionals, or anisotropy, tensor-valued functionals on convex bodies are useful. The classical approach leading to the intrinsic volumes, namely the Steiner formula for parallel bodies, can be extended by replacing the volume by higher moments of the volume. This leads, in a natural way, to a series of tensor-valued valuations. These so-called Minkowski tensors are introduced in the present chapter, and their properties are studied. A version of Hadwiger’s theorem for tensor valuations is stated. The next natural step is a localization of the Minkowski tensors, in the form of tensor-valued measures. The essential valuation, equivariance and continuity properties of these local Minkowski tensors are collected. The main goal is then a description of the vector space of all tensor valuations on convex bodies sharing these properties. Continuity properties of local Minkowski tensors and of support measures follow from continuity properties of normal cycles of convex bodies. We establish Hölder continuity of the normal cycles of convex bodies, which provides a quantitative improvement of the aforementioned continuity property.