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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $\\zeta$ be the [[Riemann zeta function|1XWBY4N0VT]].\nThen there is a [[lattice|WN9V7WSJKF]] [[sphere packing|HF32CBN5MA]] of equal balls in $\\R^n$ whose [[density|4P5PSWADJD]] is at least\n$$\\frac{\\zeta(n)}{2^{n-1}}.$$\n\n## Remarks\n\nMinkowski stated the result, and Edmund Hlawka gave the first proof in 1943.\nA standard proof averages over lattices.\nSiegel's mean value theorem says that for a random lattice $\\Lambda$ of covolume $1$ and an integrable $f$, the mean of $\\sum_{v \\in \\Lambda \\setminus \\set{0}} f(v)$ is $\\int_{\\R^n} f$.\nApplied to the indicator of a ball of volume $V \u003c 1$, this gives a lattice of covolume $1$ with no nonzero point in the ball.\nBalls of half the radius centred at its points then form a packing of density $V/2^n$.\nCounting only primitive vectors and using the symmetry $v \\mapsto -v$ improves the condition to $V \u003c 2\\zeta(n)$, which gives the bound $\\zeta(n)/2^{n-1}$.\nFor $n = 2$ the bound is $\\zeta(2)/2 = \\pi^2/12 \\approx 0.82$, below the true optimum $\\pi/\\sqrt{12} \\approx 0.91$.\nThe argument is not constructive: it shows that good lattices exist without exhibiting one.\n\n## Explanations\n\n### L1 human\n\nIn high dimensions it is hard to pack balls well, and good explicit packings are rare.\nThe theorem guarantees that some lattice arrangement fills at least about $2^{-n+1}$ of space.\nIt does so by showing that a randomly chosen lattice does this well on average, so a good one must exist.\n\n### L2 human\n\nThe lower bound $\\zeta(n) 2^{1-n}$ is roughly $2^{-n}$, and upper bounds such as [[Rogers' bound|QHPXB39SN7]] and the Kabatiansky–Levenshtein bound are also exponentially small, but with a different exponent.\nClosing this exponential gap is a central open problem.\nLater work improved the lower bound by factors linear in $n$, for example Ball's bound $2(n - 1) \\zeta(n) 2^{-n}$, and Venkatesh obtained a further $\\log \\log n$ factor in infinitely many dimensions.\nThe same averaging idea underlies existence proofs for good error-correcting codes.\n",
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