Cones and Cosmic Closure
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.
KeywordsConvex Function Convex Hull Convex Cone Homogeneous Function Convex Case
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