{
  "id": "2304.07859",
  "version": "v1",
  "published": "2023-04-16T18:49:21.000Z",
  "updated": "2023-04-16T18:49:21.000Z",
  "title": "Affine Isoperimetric Inequalities for Higher-Order Projection and Centroid Bodies",
  "authors": [
    "Julián Haddad",
    "Dylan Langharst",
    "Eli Putterman",
    "Michael Roysdon",
    "Deping Ye"
  ],
  "comment": "39-42 pages Keywords: Projection bodies, Centroid bodies, Radial mean bodies, Busemann-Petty centroid inequality, Steiner symmetrization, Petty projection inequality, affine Sobolev inequality",
  "categories": [
    "math.FA",
    "math.CA",
    "math.MG"
  ],
  "abstract": "In 1970, Schneider generalized the difference body of a convex body to higher-order, and also established the higher-order analogue of the Rogers-Shephard inequality. In this paper, we extend this idea to the projection body, centroid body, and radial mean bodies, as well as prove the associated inequalities (analogues of Zhang's projection inequality, Petty's projection inequality, the Busemann-Petty centroid inequality and Busemann's random simplex inequality). We also establish a new proof of Schneider's higher-order Rogers-Shephard inequality. As an application, a higher-order affine Sobolev inequality for functions of bounded variation is provided.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-16T18:49:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A39",
      "52A40",
      "28A75",
      "46E35"
    ],
    "keywords": [
      "affine isoperimetric inequalities",
      "centroid body",
      "higher-order projection",
      "busemanns random simplex inequality",
      "higher-order affine sobolev inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}