|
In mathematics, Shephard's problem, is the following geometrical question asked by : if ''K'' and ''L'' are centrally symmetric convex bodies in ''n''-dimensional Euclidean space such that whenever ''K'' and ''L'' are projected onto a hyperplane, the volume of the projection of ''K'' is smaller than the volume of the projection of ''L'', then does it follow that the volume of ''K'' is smaller than that of ''L''? In this case, "centrally symmetric" means that the reflection of ''K'' in the origin, ''−K'', is a translate of ''K'', and similarly for ''L''. If ''π''''k'' : R''n'' → Π''k'' is a projection of R''n'' onto some ''k''-dimensional hyperplane Π''k'' (not necessarily a coordinate hyperplane) and ''V''''k'' denotes ''k''-dimensional volume, Shephard's problem is to determine the truth or falsity of the implication : ''V''''k''(''π''''k''(''K'')) is sometimes known as the brightness of ''K'' and the function ''V''''k'' o ''π''''k'' as a (''k''-dimensional) brightness function. In dimensions ''n'' = 1 and 2, the answer to Shephard's problem is "yes". In 1967, however, Petty and Schneider showed that the answer is "no" for every ''n'' ≥ 3. The solution of Shephard's problem requires Minkowski's first inequality for convex bodies. ==See also== *Busemann–Petty 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「Shephard's problem」の詳細全文を読む スポンサード リンク
|