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  "body": "\n## Definition\n\nLet $S \\subset \\R^n$ be a nonempty set with no accumulation point, and let $\\abs{x - y}$ denote the [[Euclidean distance|0NPPGHQNXQ]].\nThe *Voronoi cell* of a point $p \\in S$ is\n$$V(p) = \\set{x \\in \\R^n : \\abs{x - p} \\le \\abs{x - q} \\text{ for all } q \\in S}.$$\nIf $\\Lambda$ is a [[lattice|WN9V7WSJKF]] in $\\R^n$, the *Voronoi cell of* $\\Lambda$ is the cell $V(0)$ of the origin in $S = \\Lambda$.\n\n## Remarks\n\nEach condition $\\abs{x - p} \\le \\abs{x - q}$ defines a closed half-space, so $V(p)$ is a closed convex set.\nExample: for $\\Lambda = \\Z^2$, the cell is $V(0) = [-\\tfrac12, \\tfrac12]^2$.\nIndeed, $\\abs{x} \\le \\abs{x - q}$ for $q = (\\pm 1, 0)$ and $q = (0, \\pm 1)$ gives $\\abs{x_1} \\le \\tfrac12$ and $\\abs{x_2} \\le \\tfrac12$, and these four conditions imply the others.\nNon-example: the closed square $[0, 1]^2$ is not the Voronoi cell of $0$ in $\\Z^2$, because its point $(1, 1)$ is at distance $0$ from $q = (1, 1)$ and at distance $\\sqrt 2$ from $0$.\nFor a lattice $\\Lambda$, the cell $V(0)$ is a convex polytope, symmetric about $0$, its translates $V(0) + v$ with $v \\in \\Lambda$ cover $\\R^n$ with disjoint interiors, and its volume equals $\\det \\Lambda$.\nThe cells of all points of $S$ form the Voronoi diagram of $S$.\n",
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