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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $\\operatorname{vol}$ be the [[Lebesgue measure|2Z2QVWZ262]] on $\\R^n$, and let $B_R$ be the [[closed ball|0Q0H13G8ZB]] of radius $R$ centred at $0$.\nA *covering* of $\\R^n$ is a countable family $\\mathcal C = (K_i)_{i \\in I}$ of compact sets whose union is $\\R^n$.\nIt is required that each bounded set meets only finitely many $K_i$.\nThe *lower density* of $\\mathcal C$ is\n$$\\underline{d}(\\mathcal C) = \\liminf_{R \\to \\infty} \\frac{\\sum_{i \\in I} \\operatorname{vol}(K_i \\cap B_R)}{\\operatorname{vol}(B_R)},$$\nwith the [[limit inferior|0H4R3RESGV]]; when the limit exists it is the *density* of $\\mathcal C$.\nLet $K$ be a compact set with nonempty interior.\nThe *covering density* of $K$ is $\\vartheta(K) = \\inf \\underline{d}(\\mathcal C)$, over all coverings $\\mathcal C$ of $\\R^n$ by congruent copies of $K$.\n\n## Remarks\n\nThe ratio is the same as for the [[packing density|4P5PSWADJD]]; packings forbid overlaps, coverings forbid gaps.\nSince the sets cover $B_R$, the ratio is at least $1$, so $\\vartheta(K) \\ge 1$, with equality when $K$ tiles space, as a cube does.\nExample: unit discs centred at the points of the triangular lattice spanned by $(\\sqrt 3, 0)$ and $(\\sqrt 3/2, 3/2)$.\nEach disc contains the regular hexagon of circumradius $1$ around its centre, and these hexagons tile the plane, so the discs cover.\nThe density is $\\pi$ divided by the hexagon area $3\\sqrt 3/2$, that is $2\\pi/\\sqrt{27}$.\nNon-example: unit discs centred at $2\\Z^2$ do not form a covering, since the point $(1, 1)$ is at distance $\\sqrt 2 \u003e 1$ from every centre.\nFor a lattice covering $K + \\Lambda$, the density exists and equals $\\operatorname{vol}(K)/\\det \\Lambda$.\n",
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