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In mathematics, Anderson's theorem is a result in real analysis and geometry which says that the integral of an integrable, symmetric, unimodal, non-negative function ''f'' over an ''n''-dimensional convex body ''K'' does not decrease if ''K'' is translated inwards towards the origin. This is a natural statement, since the graph of ''f'' can be thought of as a hill with a single peak over the origin; however, for ''n'' ≥ 2, the proof is not entirely obvious, as there may be points ''x'' of the body ''K'' where the value ''f''(''x'') is larger than at the corresponding translate of ''x''. Anderson's theorem also has an interesting application to probability theory. ==Statement of the theorem== Let ''K'' be a convex body in ''n''-dimensional Euclidean space R''n'' that is symmetric with respect to reflection in the origin, i.e. ''K'' = −''K''. Let ''f'' : R''n'' → R be a non-negative, symmetric, globally integrable function; i.e. * ''f''(''x'') ≥ 0 for all ''x'' ∈ R''n''; * ''f''(''x'') = ''f''(−''x'') for all ''x'' ∈ R''n''; * Suppose also that the super-level sets ''L''(''f'', ''t'') of ''f'', defined by : are convex subsets of R''n'' for every ''t'' ≥ 0. (This property is sometimes referred to as being unimodal.) Then, for any 0 ≤ ''c'' ≤ 1 and ''y'' ∈ R''n'', : 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「Anderson's theorem」の詳細全文を読む スポンサード リンク
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