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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 with the original body, and that furthermore, the upper bound of 2''n'' is necessary iff the body is a parallelepiped. There also exists an equivalent formulation in terms of the number of floodlights needed to illuminate the body. The Hadwiger conjecture is named after Hugo Hadwiger, who included it on a list of unsolved problems in 1957; it was, however, previously studied by and independently, . Additionally, there is a different Hadwiger conjecture concerning graph coloring—and in some sources the geometric Hadwiger conjecture is also called the Levi–Hadwiger conjecture or the Hadwiger–Levi covering problem. The conjecture remains unsolved even in three dimensions, though the two dimensional case was resolved by . ==Formal statement== Formally, the Hadwiger conjecture is: If ''K'' is any bounded convex set in the ''n''-dimensional Euclidean space R''n'', then there exists a set of 2''n'' scalars ''s''''i'' and a set of 2''n'' translation vectors ''v''''i'' such that all ''s''''i'' lie in the range 0 < ''s''''i'' < 1, and : Furthermore, the upper bound is necessary iff ''K'' is a parallelepiped, in which case all 2''n'' of the scalars may be chosen to be equal to 1/2. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hadwiger conjecture (combinatorial geometry)」の詳細全文を読む スポンサード リンク
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