{
  "id": "1903.03100",
  "version": "v1",
  "published": "2019-03-07T18:53:08.000Z",
  "updated": "2019-03-07T18:53:08.000Z",
  "title": "Characterisation of Valuations and Curvature Measures in Euclidean Spaces",
  "authors": [
    "Mykhailo Saienko"
  ],
  "comment": "23 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "Valuations constitute a large 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 $\\mathbb{R}$ were 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 $SO(n)$-covariant curvature measures with values in any $SO(n)$-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 $SO(n)$-covariant valuations with values in any $SO(n)$-representation. Furthermore, a decomposition of the space of smooth translation-invariant $\\mathbb{R}$-valued curvature measures as an $SO(n)$-representation is obtained. As a corollary, we construct an explicit basis of continuous translation-invariant $\\mathbb{R}$-valued valuations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-07T18:53:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C65",
      "53A45",
      "58A10"
    ],
    "keywords": [
      "euclidean spaces",
      "smooth translation-invariant",
      "yield local information",
      "continuous translation-invariant",
      "covariant curvature measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}