{
  "id": "2406.13581",
  "version": "v1",
  "published": "2024-06-19T14:18:28.000Z",
  "updated": "2024-06-19T14:18:28.000Z",
  "title": "Optimal constants in concentration inequalities on the sphere and in the Gauss space",
  "authors": [
    "Guillaume Aubrun",
    "Justin Jenkinson",
    "Stanislaw J. Szarek"
  ],
  "comment": "29 pages, 6 figures",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "We show several variants of concentration inequalities on the sphere stated as subgaussian estimates with optimal constants. For a Lipschitz function, we give one-sided and two-sided bounds for deviation from the median as well as from the mean. For example, we show that if $\\mu$ is the normalized surface measure on $S^{n-1}$ with $n\\geq 3$, $f : S^{n-1} \\to \\mathbb{R}$ is $1$-Lipschitz, $M$ is the median of $f$, and $t >0$, then $\\mu\\big(f \\geq M +t\\big) \\leq \\frac 12 e^{-nt^2/2}$. If $M$ is the mean of $f$, we have a two-sided bound $\\mu\\big(|f - M| \\geq t\\big) \\leq e^{-nt^2/2}$. Consequently, if $\\gamma$ is the standard Gaussian measure on $\\mathbb{R}^n$ and $f : \\mathbb{R}^{n} \\to \\mathbb{R}$ (again, $1$-Lipschitz, with the mean equal to $M$), then $\\gamma \\big(|f - M| \\geq t\\big) \\leq e^{-t^2/2}$. These bounds are slightly better and arguably more elegant than those available elsewhere in the literature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-19T14:18:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "concentration inequalities",
      "optimal constants",
      "gauss space",
      "standard gaussian measure",
      "two-sided bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}