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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\Omega \\subseteq \\R^n$ be a bounded open [[convex set|02AS45T3H9]].\nWrite $\\abs{pq}$ for the [[Euclidean distance|0NPPGHQNXQ]] of points $p$ and $q$.\nLet $x, y \\in \\Omega$ be distinct.\nThe line through $x$ and $y$ meets the boundary of $\\Omega$ in exactly two points $a$ and $b$.\nName them so that $a, x, y, b$ lie on the line in this order.\nThe *Hilbert metric* on $\\Omega$ is\n$$d_\\Omega(x, y) = \\frac{1}{2} \\log \\frac{\\abs{ay} \\, \\abs{xb}}{\\abs{ax} \\, \\abs{yb}},$$\nwith $d_\\Omega(x, x) = 0$.\n\n## Remarks\n\nThe map $d_\\Omega$ is a [[metric|063W5JMY85]] on $\\Omega$; the triangle inequality uses the convexity of $\\Omega$.\n\nExample: let $\\Omega = (-1, 1) \\subset \\R$ and $0 \u003c y \u003c 1$.\nFor $x = 0$ the boundary points are $a = -1$ and $b = 1$, so $d_\\Omega(0, y) = \\frac{1}{2} \\log \\frac{1 + y}{1 - y}$, which tends to $\\infty$ as $y \\to 1$.\n\nNon-example: the open half-plane $\\set{(u, v) \\in \\R^2 : v \u003e 0}$ is convex but not bounded.\nA horizontal line in it does not meet the boundary at all, so the formula has no meaning.\n\nSome authors omit the factor $\\frac{1}{2}$.\nWhen $\\Omega$ is the open unit ball, $d_\\Omega$ is the hyperbolic distance of the [[Beltrami–Klein model|1WN9B18H86]].\n",
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