Rotation covariant local tensor valuations on convex bodies

Abstract

For valuations on convex bodies in Euclidean spaces, there is by now a long series of characterization and classification theorems. The classical template is Hadwiger’s theorem, saying that every rigid motion invariant, continuous, real-valued valuation on convex bodies in [Formula: see text] is a linear combination of the intrinsic volumes. For tensor-valued valuations, under the assumptions of isometry covariance and continuity, there is a similar classification theorem, due to Alesker. Also for the local extensions of the intrinsic volumes, the support, curvature and area measures, there are analogous characterization results, with continuity replaced by weak continuity, and involving an additional assumption of local determination. The present authors have recently obtained a corresponding characterization result for local tensor valuations, or tensor-valued support measures (generalized curvature measures), of convex bodies in [Formula: see text]. The covariance assumed there was with respect to the group [Formula: see text] of orthogonal transformations. This was suggested by Alesker’s observation, according to which in dimensions [Formula: see text], the weaker assumption of [Formula: see text] covariance does not yield more tensor valuations. However, for tensor-valued support measures, the distinction between proper and improper rotations does make a difference. This paper considers, therefore, the local tensor valuations sharing the previously assumed properties, but with [Formula: see text] covariance replaced by [Formula: see text] covariance, and provides a complete classification. New tensor-valued support measures appear only in dimensions 2 and 3.