{
  "id": "0804.0453",
  "version": "v2",
  "published": "2008-04-03T00:29:53.000Z",
  "updated": "2008-12-24T19:51:07.000Z",
  "title": "On the role of Convexity in Functional and Isoperimetric Inequalities",
  "authors": [
    "Emanuel Milman"
  ],
  "comment": "42 pages, to appear in the Proceedings of the London Mathematical Society; small revision",
  "doi": "10.1112/plms/pdn045",
  "categories": [
    "math.FA",
    "math.PR"
  ],
  "abstract": "This is a continuation of our previous work 0712.4092. It is well known that various isoperimetric inequalities imply their functional ``counterparts'', but in general this is not an equivalence. We show that under certain convexity assumptions (e.g. for log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz-Sobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz'ya and extended by Barthe--Cattiaux--Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no Central-Limit obstruction. As another application, we show that under our convexity assumptions, $q$-log-Sobolev inequalities ($q \\in [1,2]$) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry--Ledoux and Bobkov--Zegarlinski. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the $CD(0,\\infty)$ curvature-dimension condition of Bakry--\\'Emery.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-12-24T19:51:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32F32",
      "26D10",
      "46E35",
      "31C15"
    ],
    "keywords": [
      "inequality",
      "isoperimetric inequalities",
      "convexity assumptions",
      "log-concave probability measures",
      "orlicz-sobolev functional inequalities"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.0453M"
    }
  }
}