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Cones and Cosmic Closure

Chapter
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 317)

An important advantage that the extended real line ℝ¯ has over the real line ℝ is compactness: every sequence of elements has a convergent subsequence. This property is achieved by adjoining to ℝ the special elements oo and −ø, which can act as limits for unbounded sequences under special rules. An analogous compactification is possible for ℝ n . It serves in characterizing basic ‘growth’ properties that sets and functions may have in the large.

Every vector x ≠ 0 in ℝ n has both magnitude and direction. The magnitude of x is ÀxÀ, which can be manipulated in familiar ways. The direction of x has often been underplayed as a mathematical entity, but our interest now lies in a rigorous treatment where directions are viewed as ‘points at infinity’ to be adjoined to ordinary space.

Keywords

Convex Function Convex Hull Convex Cone Homogeneous Function Convex Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

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