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  "body": "\n## Problem\n\nLet $n \\ge 1$, and let $K \\subseteq \\R^n$ be a [[convex body|437FSG3419]].\nA nonzero vector $u$ *illuminates* a [[boundary|05D5EGMFYV]] point $x$ of $K$ if the ray $\\set{x + t u : t \u003e 0}$ meets the [[interior|0RZH6W98HK]] of $K$.\nThe *illumination number* $I(K)$ is the least number of vectors such that every boundary point of $K$ is illuminated by one of them.\nShow that $I(K) \\le 2^n$, with equality only when $K$ is a parallelotope.\n\n## Remarks\n\nBoltyanski proved that $I(K)$ equals the least number of smaller homothetic copies $x_i + \\lambda K$, with $0 \u003c \\lambda \u003c 1$, that cover $K$.\nSo the problem is equivalent to the [[Hadwiger covering conjecture|SVP75YY386]], and it is open for $n \\ge 3$.\nExample: for a disc, a vector $u$ illuminates the boundary point with outer normal $\\nu$ exactly when $u \\cdot \\nu \u003c 0$.\nThree vectors at angles $2\\pi/3$ illuminate every boundary point, while two vectors never suffice, so $I = 3$.\nFor the square, each corner needs its own direction, so $I = 4 = 2^2$.\n\n## Explanations\n\n### L1 human\n\nImagine a convex solid lit by floodlights placed very far away, each shining in one fixed direction.\nA point of the surface counts as lit if the light, coming in, can pass through it into the solid.\nHow many floodlights are needed to light the whole surface?\nA round ball needs few, a cube needs one for each corner, and the conjecture says that no convex body in $n$ dimensions needs more than $2^n$.\n",
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      "first_stated": "1960",
      "posed_by": "Vladimir Boltyansky"
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