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In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces. These properties can be defined in two seemingly different ways: hyperconvexity involves the intersection properties of closed balls in the space, while injectivity involves the isometric embeddings of the space into larger spaces. However it is a theorem of Aronszajn and Panitchpakdi (1956; see e.g. Chepoi 1997) that these two different types of definitions are equivalent. == Hyperconvexity == A metric space ''X'' is said to be hyperconvex if it is convex and its closed balls have the binary Helly property. That is, #any two points ''x'' and ''y'' can be connected by the isometric image of a line segment of length equal to the distance between the points (i.e. ''X'' is a path space), and #if ''F'' is any family of closed balls :: :such that each pair of balls in ''F'' meet, then there exists a point ''x'' common to all the balls in ''F''. Equivalently, if a set of points ''pi'' and radii ''ri > 0'' satisfies ''ri'' + ''rj'' ≥ ''d''(''pi'',''pj'') for each ''i'' and ''j'', then there is a point ''q'' of the metric space that is within distance ''ri'' of each ''pi''. 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「Injective metric space」の詳細全文を読む スポンサード リンク
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