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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $S = \\set{p_1, \\dots, p_k}$ be a finite set of $k \\ge 1$ distinct points of [[Euclidean space|0XPR1E85W0]] $\\R^n$, called *sites*.\nLet $\\norm{x - y}$ denote the [[Euclidean distance|0NPPGHQNXQ]].\nThe *Voronoi cell* of the site $p_i$ is\n$$V(p_i) = \\set{x \\in \\R^n : \\norm{x - p_i} \\le \\norm{x - p_j} \\text{ for all } j = 1, \\dots, k}.$$\nThe *Voronoi diagram* of $S$ is the family of cells $V(p_1), \\dots, V(p_k)$.\n\n## Remarks\n\nExample: for the two sites $p_1 = (-1, 0)$ and $p_2 = (1, 0)$ in $\\R^2$, the condition $\\norm{x - p_1} \\le \\norm{x - p_2}$ reduces to $x_1 \\le 0$.\nSo $V(p_1)$ is the closed half-plane $x_1 \\le 0$, $V(p_2)$ is the half-plane $x_1 \\ge 0$, and they meet along the perpendicular bisector $x_1 = 0$.\n\nNon-example: the cells $\\set{x : x_1 \\le -1}$ and $\\set{x : x_1 \\ge -1}$ are not the Voronoi diagram of these two sites, since the point $(-1/2, 0)$ is closer to $p_1$ but lies in the second set only.\n\nEach condition $\\norm{x - p_i} \\le \\norm{x - p_j}$ defines a closed half-space, so each cell is a convex polyhedron.\nThe cells cover $\\R^n$, and two distinct cells have disjoint interiors, so the Voronoi diagram is a tiling of $\\R^n$.\nThe Voronoi cell of a point of a lattice, with all lattice points as sites, is a [[Voronoi cell|V1R1R88GV6]] in the sense of the geometry of numbers.\n",
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