Abstract
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, respectively. More recently, characterisation results have been achieved for curvature measures with values in Symp ℝn and Sym2Λqℝn for p, q ≥ 1 with varying assumptions as for their invariance properties.
In the present work, we classify all smooth translation-invariant SO(n)-equivariant curvature measures with values in any SO(n)-representation in terms of certain differential forms on the sphere bundle Sℝn and describe their behaviour under the globalisation map. The latter result also yields a similar classification of all continuous SO(n)-equivariant 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.
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Supported by DFG grants BE 2484/5-1 and BE 2484/5-2.
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Saienko, M. Characterisation of valuations and curvature measures in euclidean spaces. Isr. J. Math. 250, 463–500 (2022). https://doi.org/10.1007/s11856-022-2343-1
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DOI: https://doi.org/10.1007/s11856-022-2343-1