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  "body": "\n## Statement\n\nLet $d \\ge 1$ and let $P \\subset \\R^d$ be a [[parallelohedron|MR2JWBCZQZ]].\nThen there are a [[lattice|WN9V7WSJKF]] $\\Lambda \\subset \\R^d$ and an affine bijection $A \\colon \\R^d \\to \\R^d$ such that $A(P)$ is the [[Voronoi cell|V1R1R88GV6]] of $\\Lambda$.\n\n## Remarks\n\nThe converse holds: the Voronoi cell of every lattice is a parallelohedron.\nVoronoi proved the conjecture for primitive parallelohedra, those whose tiling has exactly $d + 1$ tiles at each vertex.\nDelone proved it in dimensions $d \\le 4$.\nErdahl proved it for parallelohedra that are zonotopes.\nThe general case is open.\n\n## Explanations\n\n### L1 human\n\nA parallelohedron is a convex tile that fills space when you only slide copies of it.\nEvery lattice gives such a tile: the region of points nearer to one lattice point than to all others.\nVoronoi guessed that this is the only source of such tiles, once you allow stretching and shearing.\n",
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      "first_stated": "1908",
      "posed_by": "Georgy Voronoy"
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      "52B11",
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    "name": "Voronoi's conjecture on parallelohedra",
    "sources": [
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        "url": "https://en.wikipedia.org/wiki/Parallelohedron"
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  "msc": [
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  "name": "Voronoi's conjecture on parallelohedra",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/F03/F03A4MKA5V.md",
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