{
  "id": "math/0608681",
  "version": "v2",
  "published": "2006-08-28T09:28:53.000Z",
  "updated": "2006-11-20T09:30:37.000Z",
  "title": "Modified log-Sobolev inequalities and isoperimetry",
  "authors": [
    "Alexander V. Kolesnikov"
  ],
  "comment": "26 pages",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "We find sufficient conditions for a probability measure $\\mu$ to satisfy an inequality of the type $$ \\int_{\\R^d} f^2 F\\Bigl(\\frac{f^2}{\\int_{\\R^d} f^2 d \\mu} \\Bigr) d \\mu \\le C \\int_{\\R^d} f^2 c^{*}\\Bigl(\\frac{|\\nabla f|}{|f|} \\Bigr) d \\mu + B \\int_{\\R^d} f^2 d \\mu, $$ where $F$ is concave and $c$ (a cost function) is convex. We show that under broad assumptions on $c$ and $F$ the above inequality holds if for some $\\delta>0$ and $\\epsilon>0$ one has $$ \\int_{0}^{\\epsilon} \\Phi\\Bigl(\\delta c\\Bigl[\\frac{t F(\\frac{1}{t})}{{\\mathcal I}_{\\mu}(t)} \\Bigr] \\Bigr) dt < \\infty, $$ where ${\\mathcal I}_{\\mu}$ is the isoperimetric function of $\\mu$ and $\\Phi = (y F(y) -y)^{*}$. In a partial case $${\\mathcal I}_{\\mu}(t) \\ge k t \\phi ^{1-\\frac{1}{\\alpha}} (1/t), $$ where $\\phi$ is a concave function growing not faster than $\\log$, $k>0$, $1 < \\alpha \\le 2$ and $t \\le 1/2$, we establish a family of tight inequalities interpolating between the $F$-Sobolev and modified inequalities of log-Sobolev type. A basic example is given by convex measures satisfying certain integrability assumptions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-11-20T09:30:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "26D10"
    ],
    "keywords": [
      "modified log-sobolev inequalities",
      "inequality",
      "isoperimetry",
      "basic example",
      "log-sobolev type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8681K"
    }
  }
}