Hadwiger conjecture (combinatorial geometry): Difference between revisions

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===Alternate formulation with illumination===

As shown by [[Vladimir Boltyansky|Boltyansky]], the problem is equivalent to one of illumination: how many floodlights must be placed outside of an opaque convex body in order to completely illuminate its exterior? For the purposes of this problem, a body is only considered to be illuminated if for each point of itsthe [[tangentboundary plane]]s separatesof the body, fromthere is at least one floodlight that is separated from the body by all of the floodlights[[tangent plane]]s intersecting the body on this point; thus, although the faces of a cube may be lit by only two floodlights, the planes tangent to its vertices and edges cause it to need many more lights in order for it to be fully illuminated. For any convex body, the number of floodlights needed to completely illuminate it turns out to equal the number of smaller copies of the body that are needed to cover it.<ref name="BMP"/>

==Examples==