{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $(K_m)_{m \\ge 1}$ be a sequence of nonempty [[compact|0R6EWFKRCF]] [[convex sets|02AS45T3H9]] in $\\R^n$, all contained in one [[bounded set|09QSZVVM26]].\nThen $(K_m)$ has a [[subsequence|0F1PR74BRX]] that converges, in the [[Hausdorff distance|F79WYYC07C]], to a nonempty compact convex set $K \\subseteq \\R^n$.\n\n## Remarks\n\nThe nonempty compact subsets of a compact metric space form a compact metric space under the Hausdorff distance.\nAll $K_m$ lie in a closed ball, which is compact, so a subsequence converges to some nonempty compact set $K$.\nA Hausdorff limit of convex sets is convex, which gives the theorem.\nThe limit can be smaller in dimension than the $K_m$: the rectangles $K_m = [0, 1] \\times [0, \\frac1m]$ converge to the segment $[0,1] \\times \\set{0}$.\nBoundedness is needed: the segments $[0, m] \\times \\set{0}$ have no convergent subsequence, since their Hausdorff distances to any fixed compact set tend to infinity.\n\n## Explanations\n\n### L1 human\n\nTake infinitely many convex shapes that all fit in one box.\nThen one can pick some of them, one after another, that settle down to a limiting convex shape.\nThis is the Bolzano–Weierstrass theorem for convex shapes instead of numbers.\n\n### L2 human\n\nThe theorem makes the space of convex bodies with the Hausdorff metric locally compact.\nIt is used to prove the existence of extremal bodies.\nA continuous functional, such as volume or [[mixed volume|QB3Q5FBJ08]], attains its maximum and minimum over a bounded closed family of convex sets.\nExistence proofs for the [[John ellipsoid|WNHXE7FCG4]] and for solutions of isoperimetric-type problems use it in this way.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "02AS45T3H9",
      "F79WYYC07C"
    ],
    "kind": "statement",
    "msc": [
      "52A20",
      "54B20"
    ],
    "name": "Blaschke selection theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q784049",
      "wikipedia": "Blaschke selection theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Blaschke_selection_theorem"
      }
    ],
    "tag": "D9C581V5HG",
    "uses": [
      "02AS45T3H9",
      "0R6EWFKRCF",
      "09QSZVVM26",
      "0F1PR74BRX",
      "F79WYYC07C"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/D9C581V5HG/",
  "links": [
    {
      "field": "about",
      "target": "02AS45T3H9"
    },
    {
      "field": "about",
      "target": "F79WYYC07C"
    },
    {
      "field": "uses",
      "target": "02AS45T3H9"
    },
    {
      "field": "uses",
      "target": "0R6EWFKRCF"
    },
    {
      "field": "uses",
      "target": "09QSZVVM26"
    },
    {
      "field": "uses",
      "target": "0F1PR74BRX"
    },
    {
      "field": "uses",
      "target": "F79WYYC07C"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/D9C581V5HG.md",
  "msc": [
    "52A20",
    "54B20"
  ],
  "name": "Blaschke selection theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/D9C/D9C581V5HG.md",
  "status": null,
  "tag": "D9C581V5HG",
  "uri": "m2://D9C581V5HG",
  "url": "https://tamnd.github.io/math2/wiki/blaschke-selection-theorem/",
  "writing": {
    "tag": "D9C581V5HG",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}