{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "YGSKZFTEX1"
    }
  ],
  "body": "\n## Statement\n\nLet $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $n \\ge 2$, and let $k$ be a real number.\nAssume that the [[Ricci curvature|4GTNM0WHDH]] satisfies $\\operatorname{Ric}(v, v) \\ge (n-1) k \\, g(v, v)$ for every tangent vector $v$.\nFor $r \u003e 0$ let $V_k(r)$ be the volume of a ball of radius $r$ in the simply connected $n$-dimensional [[space form|1VM1AQT956]] of curvature $k$.\nLet $p \\in M$ and let $B(p, r)$ be the ball of radius $r$ about $p$ for the [[Riemannian distance|B4QR306D8G]].\nLet $R = \\pi/\\sqrt{k}$ if $k \u003e 0$ and $R = \\infty$ if $k \\le 0$.\nThen the function\n$$r \\mapsto \\frac{\\operatorname{vol} B(p, r)}{V_k(r)}$$\nis nonincreasing on $(0, R)$.\nSince it tends to $1$ as $r \\to 0$, it follows that $\\operatorname{vol} B(p, r) \\le V_k(r)$ for $0 \u003c r \u003c R$.\n\n## Remarks\n\nThe idea of the proof: write the volume in geodesic polar coordinates about $p$, with density $\\mathcal{A}(t, \\theta)$ along the geodesic in direction $\\theta$, set to $0$ beyond the cut locus.\nThe Ricci bound and the Riccati equation for the mean curvature of distance spheres give that $\\mathcal{A}(t, \\theta) / \\bar{\\mathcal{A}}_k(t)$ is nonincreasing in $t$, where $\\bar{\\mathcal{A}}_k$ is the density of the model.\nIntegrating over $\\theta$ and $t$ gives the monotonicity of the ratio of volumes.\nThe inequality $\\operatorname{vol} B(p, r) \\le V_k(r)$ is due to Bishop; the monotonicity is due to Gromov.\n\nExample: for $\\operatorname{Ric} \\ge 0$ the comparison is with $\\R^n$, so $\\operatorname{vol} B(p, r) / (\\omega_n r^n)$ is nonincreasing, where $\\omega_n$ is the volume of the unit ball of $\\R^n$.\nFor $M = \\R^n$ the ratio is constant equal to $1$.\n\n## Explanations\n\n### L1 human\n\nA lower bound on Ricci curvature means volumes grow no faster than in the model space.\nBalls in $M$ are never larger than balls of the same radius in the model, and the ratio of the two volumes can only decrease as the radius grows.\n\n### L2 human\n\nThe inequality is the basic volume estimate under a Ricci lower bound.\nIts monotonicity gives the doubling property $\\operatorname{vol} B(p, 2r) \\le C \\operatorname{vol} B(p, r)$ with $C$ depending only on $n$, $k$ and an upper bound for $r$.\nThis is the input for Gromov's precompactness theorem for the Gromov–Hausdorff distance and for many estimates in geometric analysis.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "4GTNM0WHDH"
    ],
    "kind": "statement",
    "msc": [
      "53C21",
      "53C20",
      "53C23"
    ],
    "name": "Bishop–Gromov inequality",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q4918129",
      "wikipedia": "Bishop–Gromov inequality"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bishop%E2%80%93Gromov_inequality"
      }
    ],
    "tag": "82F5Q4F1S6",
    "uses": [
      "7CBT4SM83Y",
      "4GTNM0WHDH",
      "B4QR306D8G",
      "1VM1AQT956"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/82F5Q4F1S6/",
  "links": [
    {
      "field": "about",
      "target": "4GTNM0WHDH"
    },
    {
      "field": "uses",
      "target": "7CBT4SM83Y"
    },
    {
      "field": "uses",
      "target": "4GTNM0WHDH"
    },
    {
      "field": "uses",
      "target": "B4QR306D8G"
    },
    {
      "field": "uses",
      "target": "1VM1AQT956"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/82F5Q4F1S6.md",
  "msc": [
    "53C21",
    "53C20",
    "53C23"
  ],
  "name": "Bishop–Gromov inequality",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/82F/82F5Q4F1S6.md",
  "status": null,
  "tag": "82F5Q4F1S6",
  "uri": "m2://82F5Q4F1S6",
  "url": "https://math2.tamnd.com/wiki/bishop-gromov-inequality/",
  "writing": {
    "tag": "82F5Q4F1S6",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}