{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]].\nFor points $x, y, z \\in X$, say that $y$ lies *between* $x$ and $z$ when $y \\ne x$, $y \\ne z$ and $d(x, y) + d(y, z) = d(x, z)$.\nThe space $X$ is a *Busemann G-space* when it satisfies these four conditions.\n\n1. Finite compactness: every bounded closed subset of $X$ is compact.\n2. Menger convexity: for all distinct $x, z \\in X$ there is a point between $x$ and $z$.\n3. Local prolongability: each $p \\in X$ has a radius $r_p \u003e 0$ with this property. For distinct $x, y$ in the [[open ball|0DC3MPR7C9]] $B(p, r_p)$, some $z \\in X$ has $y$ between $x$ and $z$.\n4. Uniqueness of prolongation: if $y$ lies between $x$ and $z_1$ and between $x$ and $z_2$, and $d(y, z_1) = d(y, z_2)$, then $z_1 = z_2$.\n\n## Remarks\n\nCondition 1 says that $X$ is a [[proper metric space|0Z4V98VN5E]].\nExample: the Euclidean plane $\\R^2$.\nIt is proper, the midpoint lies between any two distinct points, and a segment from $x$ through $y$ extends along its line.\nThe extension of length $t$ beyond $y$ is unique, since equality in the triangle inequality forces $z$ onto the ray from $x$ through $y$.\n\nNon-example: $\\R^2$ with the taxicab distance $d_1$ fails condition 4.\nTake $x = (0, 0)$, $y = (1, 0)$, $z_1 = (2, 0)$ and $z_2 = (\\frac{3}{2}, \\frac{1}{2})$.\nThen $d_1(y, z_1) = d_1(y, z_2) = 1$ and $d_1(x, z_1) = d_1(x, z_2) = 2$, so $y$ lies between $x$ and both points, but $z_1 \\ne z_2$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "EKXQFD7EMF",
    "kind": "definition",
    "msc": [
      "53C70",
      "51F99"
    ],
    "name": "Busemann G-space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Busemann_G-space"
      }
    ],
    "tag": "1EHBR0VMH3",
    "uses": [
      "063W5JMY85",
      "0Z4V98VN5E",
      "0DC3MPR7C9"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/1EHBR0VMH3/",
  "links": [
    {
      "field": "defines",
      "target": "EKXQFD7EMF"
    },
    {
      "field": "uses",
      "target": "063W5JMY85"
    },
    {
      "field": "uses",
      "target": "0Z4V98VN5E"
    },
    {
      "field": "uses",
      "target": "0DC3MPR7C9"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/1EHBR0VMH3.md",
  "msc": [
    "53C70",
    "51F99"
  ],
  "name": "Busemann G-space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1EH/1EHBR0VMH3.md",
  "status": null,
  "tag": "1EHBR0VMH3",
  "uri": "m2://1EHBR0VMH3",
  "url": "https://tamnd.github.io/math2/wiki/busemann-g-space-definition/",
  "writing": {
    "tag": "1EHBR0VMH3",
    "kind": "definition",
    "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
      }
    ]
  }
}