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In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the ''n''-dimensional complex space C''n''. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy. Let : is a relatively compact subset of for all real numbers In other words, a domain is pseudoconvex if has a continuous plurisubharmonic exhaustion function. Every (geometrically) convex set is pseudoconvex. When has a (twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. More specifically, with a boundary, it can be shown that has a defining function; i.e., that there exists which is so that , and . Now, is pseudoconvex iff for every and in the complex tangent space at p, that is, :, we have : If does not have a boundary, the following approximation result can come in useful. Proposition 1 ''If is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains with (smooth) boundary which are relatively compact in , such that'' : This is because once we have a as in the definition we can actually find a ''C''∞ exhaustion function. ==The case ''n'' = 1== In one complex dimension, every open domain is pseudoconvex. The concept of pseudoconvexity is thus more useful in dimensions higher than 1. 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「pseudoconvexity」の詳細全文を読む スポンサード リンク
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