Moments and valuations

Abstract

All measurable and SL(n)-covariant vector valued valuations on convex polytopes containing the origin in their interiors are completely classified. The moment vector is shown to be essentially the only such valuation.Mathematics subject classification: 52A20, 52B45

Centro-affine tensor valuations

“…In both papers, she assumed compatibility with the whole general linear group. A version for the vector valued case that only assumes compatibility with the special linear group was very recently proved by the authors [18]. The present article is the first one to establish a classification for tensor valuations of arbitrary rank in the context of "centro-affine geometry".…”

“…The new techniques developed in this paper enabled him to prove a long sought after characterization of the rigid motion compatible Minkowski tensors in [2]. In recent years, tensor valuations were studied intensively (see, e.g., [3,5,18,21,22,25,28,46]). This is in part due to their applications in morphology and anisotropy analysis of cellular, granular or porous structures (see, e.g., [4,[41][42][43]).…”

Valuations on Convex Functions

“…Recall, that as a convex function v ω is locally Lipschitz and differentiable almost everywhere on the interior of its domain. Using polar coordinates, (14) and the substitution v ω (r) = s, we obtain from (17) that…”

Tensor valuations on lattice polytopes

“…For more information, see [43]. The dissection (18) is also used in the proof of the following result.…”