Characterisation of Valuations and Curvature Measures in Euclidean Spaces [article]

Mykhailo Saienko
2020 arXiv   pre-print
Valuations constitute a class of functionals on convex bodies which include the Euler-characteristic, the surface area, the Lebesgue-measure, and many more classical functionals. Curvature measures may be regarded as "localised" versions of valuations which yield local information about the geometry of a body's boundary. A complete classification of continuous translation-invariant SO(n)-invariant valuations and curvature measures with values in ℝ was obtained by Hadwiger and Schneider,
more » ... vely. More recently, characterisation results have been achieved for curvature measures with values in Sym^p ℝ^n and Sym^2Λ^qℝ^n for p,q ≥ 1 with varying assumptions as for their invariance properties. In the present work, we classify all smooth translation-invariant SO(n)-covariant curvature measures with values in any SO(n)-representation in terms of certain differential forms on the sphere bundle Sℝ^n and describe their behaviour under the globalisation map. The latter result also yields a similar classification of all continuous SO(n)-covariant valuations with values in any SO(n)-representation. Furthermore, a decomposition of the space of smooth translation-invariant ℝ-valued curvature measures as an SO(n)-representation is obtained. As a corollary, we construct an explicit basis of continuous translation-invariant ℝ-valued valuations.
arXiv:1903.03100v2 fatcat:sshdluekubeg7d3fnh4cld5t64