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The analyst's traveling salesman problem is an analog of the traveling salesman problem in combinatorial optimization. In its simplest and original form, it asks under what conditions may a set ''E'' in two-dimensional Euclidean space be contained inside a rectifiable curve of finite length. So while in the original traveling salesman problem, one asks for the shortest way to visit every vertex in a graph with a discrete path, this analytical version requires the curve to visit perhaps infinitely many points. ==β-numbers== A posteriori, for ''E'' to be contained in a rectifiable curve Γ, since Γ has tangents at ''H''1-almost every point in Γ (where ''H''1 denotes one-dimensional Hausdorff measure), ''E'' must look ''flat'' when you zoom in on points in ''E''. This suggests that a condition that would tell us whether a set could be contained in a curve must somehow incorporate information about how flat ''E'' is when we zoom in on points of ''E'' at different scales. This discussion motivates the definition of the following quantity: : Where ''Q'' is any square, is the sidelength of ''Q'', and dist(''x'', ''L'') measures the distance from ''x'' to the line ''L''. Intuitively, is the width of the smallest rectangle containing the portion of ''E'' inside ''Q'', and hence gives us a scale invariant notion of ''flatness''. 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「analyst's traveling salesman theorem」の詳細全文を読む スポンサード リンク
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