Hadwiger conjecture (combinatorial geometry): Difference between revisions

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Latest revision as of 21:53, 20 August 2025

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{{see also|Hadwiger conjecture (graph theory)}}

[[File:Hadwiger covering.svg|thumb|300px|A triangle can be covered by three smaller copies of itself; a square requires four smaller copies]]

{{unsolved|mathematics|Can every <math>n</math>-dimensional convex body be covered by <math>2^n</math> smaller copies of itself?}}

In [[combinatorial geometry]], the '''Hadwiger conjecture''' states that any [[convex body]] in ''n''-dimensional [[Euclidean space]] can be covered by 2^''n'' or fewer smaller bodies [[Homothetic transformation|homothetic]] with the original body, and that furthermore, the upper bound of 2^''n'' is necessary if and only if the body is a [[parallelepiped]]. There also exists an equivalent formulation in terms of the number of floodlights needed to illuminate the body.