Volumetric Discrete Geometry by Langi Zsolt; Bezdek Károly; Bezdek Karoly

Author:Langi, Zsolt; Bezdek, Károly; Bezdek, Karoly
Language: eng
Format: epub
Publisher: CRC Press LLC
Published: 2019-07-15T00:00:00+00:00

By the condition (1.3) and the relation (6.19), we have that this quantity is negative, implying that z(p) is negative for all p ∈ Dt if t is sufficiently large. Let t′ be chosen to satisfy this property. Without loss of generality, we may also assume that Xt does not intersect the hyperplane H. Let denote the set of points in D′ with xd+1-coordinates less than t′, and let V = ℍd+1\ . Then V is a neighborhood of q in ℍd+1, contained in U, and V has the property that the integral curve through any boundary point p of V leaves V at p. This proves (ii).

Now we finish the proof of Theorem 29. By the conditions in the formulation of the theorem, the set intCi ⊂ is disconnected. Let the components of this set be X1, X2, …, Xr. By Lemma 159, the integral curve of every point p ∈ D terminates at some point of these sets. Let Yj denote the points of D whose integral curve ends at a point of Xj. By Lemma 159, no Yj is empty, and it also implies that Yj is open in D for all js. Thus, D is the disjoint union of the r open sets Y1, Y2, …, Yr, where r > 1. On the other hand, D is an open convex set, and thus, it is connected; a contradiction.

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