{
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  "body": "\n## Problem\n\nLet $d \\ge 2$.\nA set $D \\subseteq \\R^d$ is a *Danzer set* if it meets every [[convex body|437FSG3419]] in $\\R^d$ of volume $1$, where volume is [[Lebesgue measure|2Z2QVWZ262]].\nIs there a Danzer set $D$ of bounded density?\nThis means that there is a constant $C$ such that every ball of radius $r \\ge 1$ contains at most $C r^d$ points of $D$.\n\n## Remarks\n\nLudwig Danzer asked the question; it is listed as a problem of Danzer in the proceedings of a convexity colloquium held in Copenhagen in 1965.\nThe question is open.\nBambah and Woods showed in 1971 that a finite union of lattices is never a Danzer set.\nThey also constructed Danzer sets with at most $C r^d \\log^{d-1} r$ points in a ball of radius $r$, by overlaying rectangular grids of different shapes.\nSolomon and Weiss gave a construction with $O(r^d \\log r)$ points, and they showed that the vertex sets of substitution tilings, such as Penrose tilings, are not Danzer sets.\n\n## Explanations\n\n### L1 human\n\nCan one scatter points in the plane, not too densely, so that every convex shape of area one contains at least one of them?\nVery long and thin shapes are the difficulty: they can slip between the points of any grid.\nAdding a little more than a constant density of points is known to be enough.\nWhether bounded density suffices is open.\n",
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      "first_stated": "1965",
      "posed_by": "Ludwig Danzer"
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    "kind": "problem",
    "msc": [
      "52C17",
      "52A37",
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    "name": "Danzer set",
    "same_as": {
      "wikidata": "Q65121675",
      "wikipedia": "Danzer set"
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    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Danzer_set"
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    "tag": "MMHWS2TC2A",
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  "link": "https://tamnd.github.io/math2/node/MMHWS2TC2A/",
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  "msc": [
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  "name": "Danzer set",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/MMH/MMHWS2TC2A.md",
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  "tag": "MMHWS2TC2A",
  "uri": "m2://MMHWS2TC2A",
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    "tag": "MMHWS2TC2A",
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