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51: １３２人目の素数さん [] 2025/01/07(火) 22:23:17.33 ID:+fDYIL0R >>49 >擬凸集合（英: pseudoconvex set）は n 次元複素空間 Cn 内のある特殊なタイプの開集合である。 擬凸集合（英: pseudoconvex set）は n 次元複素空間 Cn 内のある特殊なタイプの開集合をモデルとして導入された、凸性に似た幾何学的条件で定義される複素多様体上の領域である。 http://rio2016.5ch.net/test/read.cgi/math/1732909516/51

53: １３２人目の素数さん [] 2025/01/08(水) 11:37:28.64 ID:Tq8fsyAE >>51 >擬凸集合（英: pseudoconvex set）は n 次元複素空間 Cn 内のある特殊なタイプの開集合をモデルとして導入された、凸性に似た幾何学的条件で定義される複素多様体上の領域である。 なるほど こういうときは、en.wikipediaを見るのが定石でして なるほど、”Every (geometrically) convex set is pseudoconvex.” C2 (twice continuously differentiable) boundary Now, G is pseudoconvex iff for every p∈∂G and w in the complex tangent space at p, that is, ∇ρ(p)w= Σi=1〜n ∂ρ(p)/∂zi wi = 0, we have ?i,j=1〜n ∂2 ρ(p)/∂zj∂¯zj wiw¯j ≧ 0 . The definition above is analogous to definitions of convexity in Real Analysis. か・・・ (参考) en.wikipedia.org/wiki/Pseudoconvexity Pseudoconvexity 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 Cn. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy. Let G⊂Cn be a domain, that is, an open connected subset. One says that G is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function φ on G such that the set {z∈G∣φ(z)<x} is a relatively compact subset of G for all real numbers x. In other words, a domain is pseudoconvex if G has a continuous plurisubharmonic exhaustion function. Every (geometrically) convex set is pseudoconvex. However, there are pseudoconvex domains which are not geometrically convex. When G has a C2 (twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. More specifically, with a C2 boundary, it can be shown that G has a defining function, i.e., that there exists ρ:Cn→R which is C2 so that G={ρ<0}, and ∂G={ρ=0}. Now, G is pseudoconvex iff for every p∈∂G and w in the complex tangent space at p, that is, ∇ρ(p)w= Σi=1〜n ∂ρ(p)/∂zi wi = 0, we have ?i,j=1〜n ∂2 ρ(p)/∂zj∂¯zj wiw¯j ≧ 0 . The definition above is analogous to definitions of convexity in Real Analysis. If G does not have a C2 boundary, the following approximation result can be useful. Proposition 1 略す http://rio2016.5ch.net/test/read.cgi/math/1732909516/53

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