{
  "id": "2302.04347",
  "version": "v1",
  "published": "2023-02-08T21:29:28.000Z",
  "updated": "2023-02-08T21:29:28.000Z",
  "title": "Inequalities for sections and projections of convex bodies",
  "authors": [
    "Apostolos Giannopoulos",
    "Alexander Koldobsky",
    "Artem Zvavitch"
  ],
  "categories": [
    "math.FA",
    "math.MG"
  ],
  "abstract": "This article belongs to the area of geometric tomography, which is the study of geometric properties of solids based on data about their sections and projections. We describe a new direction in geometric tomography where different volumetric results are considered in a more general setting, with volume replaced by an arbitrary measure. Surprisingly, such a general approach works for a number of volumetric results. In particular, we discuss the Busemann-Petty problem on sections of convex bodies for arbitrary measures and the slicing problem for arbitrary measures. We present generalizations of these questions to the case of functions. A number of generalizations of questions related to projections, such as the problem of Shephard, are also discussed as well as some questions in discrete tomography.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-08T21:29:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A20",
      "53A15",
      "52B10"
    ],
    "keywords": [
      "convex bodies",
      "arbitrary measure",
      "projections",
      "volumetric results",
      "inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}