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Cones and Cosmic Closure

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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 ℝn. It serves in characterizing basic ‘growth’ properties that sets and functions may have in the large.

Every vector x ≠ 0 in ℝn 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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© 1998 Springer-Verlag Berlin Heidelberg

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(1998). Cones and Cosmic Closure. In: Variational Analysis. Grundlehren der mathematischen Wissenschaften, vol 317. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02431-3_3

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  • DOI: https://doi.org/10.1007/978-3-642-02431-3_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62772-2

  • Online ISBN: 978-3-642-02431-3

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