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Chapter 1: First Acquaintance with Convex Sets

Chapter 1: First Acquaintance with Convex Sets

pp. 3-27

Authors

, Carnegie Mellon University, Pennsylvania, , Georgia Institute of Technology
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Extract

In this chapter, we (a) present the definition and basic examples (linear and affine subspace, polyhedron, norm ball, simplex, cone) of convex sets, (b) introduce and study the notions of convex combination and convex hull, (c) outline basic convexity-preserving operations with sets, (d) establish nonemptiness of the relative interior of a nonempty convex set and the fact that the relative interior of a convex sets is dense in its closure, and (e) introduce the notions of conic and perspective transforms of a convex set.

Keywords

  • convex set
  • convex combination
  • convex hull
  • norm ball
  • simplex
  • cone
  • conic combination
  • conic hull
  • relative interior
  • conic transform
  • perspective transform

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