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In convex geometry, the projection body Π''K'' of a centrally symmetric body ''K'' in Euclidean space is the star body such that for any vector ''u'', the support function of Π''K'' in the direction ''u'' is the (''n'' – 1)-dimensional volume of the projection of ''K'' onto the hyperplane ''u''⊥. The intersection body of ''K'' is defined similarly, as the star body such that for any vector ''u'' the radial function of ''IK'' from the origin in direction ''u'' is the (''n'' – 1)-dimensional volume of the intersection of ''K'' with the hyperplane ''u''⊥. Projection bodies were discussed by , and intersection bodies were introduced by . Minkowski showed that the projection body of a convex body is convex. showed that a centrally symmetric star-shaped body is an intersection body if and only if the function 1/||''x''|| is a positive definite distribution, where ||''x''|| is the homogeneous function of degree 1 that is 1 on the boundary of the body, and used this to show that the unit balls l, 1 < ''p'' ≤ ∞ in ''n''-dimensional space with the l''p'' norm are intersection bodies for ''n''=4 but are not intersection bodies for ''n'' ≥ 5. ==See also== *Busemann–Petty problem *Shephard's problem 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「Projection body」の詳細全文を読む スポンサード リンク
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