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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\operatorname{vol}$ be the [[Lebesgue measure|2Z2QVWZ262]] on $\\R^n$.\nLet $B_R$ be the [[closed ball|0Q0H13G8ZB]] of radius $R$ centred at $0$.\nA *packing* in $\\R^n$ is a countable family $\\mathcal P = (K_i)_{i \\in I}$ of compact sets with pairwise disjoint interiors.\nThe *upper density* of $\\mathcal P$ is\n$$\\delta(\\mathcal P) = \\limsup_{R \\to \\infty} \\frac{\\sum_{i \\in I} \\operatorname{vol}(K_i \\cap B_R)}{\\operatorname{vol}(B_R)},$$\nwith the [[limit superior|0T3RZQXDYD]]; when the limit exists it is the *density* of $\\mathcal P$.\nFor a compact set $K$ with nonempty interior, the *packing density* of $K$ is $\\delta(K) = \\sup \\delta(\\mathcal P)$, over all packings $\\mathcal P$ of congruent copies of $K$.\n\n## Remarks\n\nThe value lies in $[0, 1]$.\nExample: the packing of discs of radius $\\tfrac12$ centred at the points of $\\Z^2$.\nEach disc lies in its unit square, the squares tile the plane, so the ratio tends to $\\pi/4$, the share of each square covered by its disc.\nNon-example: discs of radius $1$ centred at $\\Z^2$ do not form a packing, since neighbouring discs overlap in a set of positive area.\nFor them the same ratio tends to $\\pi \u003e 1$.\nFor a lattice packing $K + \\Lambda$ of a body $K$, the density exists and equals $\\operatorname{vol}(K)/\\det \\Lambda$.\nRestricting to lattice packings gives the lattice packing density $\\delta_L(K) \\le \\delta(K)$.\n",
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