Characterisation of Valuations and Curvature Measures in Euclidean Spaces

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 $\mathrm{SO}(n)$-invariant valuations and curvature measures with values in $\mathbb{R}$ was obtained by Hadwiger and Schneider, respectively. More recently, characterisation results have been achieved for curvature measures with values in $\operatorname{Sym}^p \mathbb{R}^n$ and $\operatorname{Sym}^2\!\Lambda^{q} \mathbb{R}^n$ for $p,q \geq 1$ with varying assumptions as for their invariance properties. In the present work, we classify all smooth translation-invariant $\mathrm{SO}(n)$-covariant curvature measures with values in any $\mathrm{SO}(n)$-representation in terms of certain differential forms on the sphere bundle $S\mathbb{R}^n$ and describe their behaviour under the globalisation map. The latter result also yields a similar classification of all continuous $\mathrm{SO}(n)$-covariant valuations with values in any $\mathrm{SO}(n)$-representation. Furthermore, a decomposition of the space of smooth translation-invariant $\mathbb{R}$-valued curvature measures as an $\mathrm{SO}(n)$-representation is obtained. As a corollary, we construct an explicit basis of continuous translation-invariant $\mathbb{R}$-valued valuations.