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1 results

  • 1 - First Acquaintance with Convex Sets

  • from Part I - Convex sets in Rn: From First Acquaintance to Linear Programming Duality
  • Fatma Kılınç-Karzan, Carnegie Mellon University, Pennsylvania, Arkadi Nemirovski, Georgia Institute of Technology
  • Book:
    Essential Mathematics for Convex Optimization
    Published online:
    22 October 2025
    Print publication:
    26 June 2025, pp 3-27

  • Summary

    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.