14 research outputs found
Riemannian curvature measures
A famous theorem of Weyl states that if is a compact submanifold of
euclidean space, then the volumes of small tubes about are given by a
polynomial in the radius , with coefficients that are expressible as
integrals of certain scalar invariants of the curvature tensor of with
respect to the induced metric. It is natural to interpret this phenomenon in
terms of curvature measures and smooth valuations, in the sense of Alesker,
canonically associated to the Riemannian structure of . This perspective
yields a fundamental new structure in Riemannian geometry, in the form of a
certain abstract module over the polynomial algebra that
reflects the behavior of Alesker multiplication. This module encodes a key
piece of the array of kinematic formulas of any Riemannian manifold on which a
group of isometries acts transitively on the sphere bundle. We illustrate this
principle in precise terms in the case where is a complex space form.Comment: Corrected version, to appear in GAF
Valuations in affine convex geometry
In convex geometry, the constructions that assign to a convex body its
difference body, projection body, or volume have the following properties: They
are (1) invariant under volume-preserving linear changes of coordinates; (2)
continuous; and (3) finitely additive. In this paper we explore the question
whether there exist other constructions with these properties. We discover a
surprising dichotomy: There are no new examples if one assumes translation
invariance, but a plethora of examples without this assumption
The Fourier transform on valuations is the Fourier transform
Alesker has proved the existence of a remarkable isomorphism of the space of
translation-invariant smooth valuations that has the same functorial properties
as the classical Fourier transform. In this paper, we show how to directly
describe this isomorphism in terms of the Fourier transform on functions. As a
consequence, we obtain simple proofs of the main properties of the
Alesker--Fourier transform. One of these properties was previously only
conjectured by Alesker and is proved here for the first time.Comment: 32 page
On the inverse Klain map
The continuity of the inverse Klain map is investigated and the class of
centrally symmetric convex bodies at which every valuation depends continuously
on its Klain function is characterized. Among several applications, it is shown
that McMullen's decomposition is not possible in the class of
translation-invariant, continuous, positive valuations. This implies that there
exists no McMullen decomposition for translation-invariant, continuous
Minkowski valuations, which solves a problem first posed by Schneider and
Schuster.Comment: 21 pages; typos fixe