Conjugate Duality

Bruce C. Dieffenbach ()
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Bruce C. Dieffenbach: Independent author

Chapter 4 in Conjugate Duality in Economic Analysis, 2026, pp 33-38 from Springer

Abstract: Abstract Fenchel proves the conjugate-duality theorem: for a proper, closed, convex function, f**=f. That a closed, convex set is the intersection of its containing, closed, half spaces is key to proving conjugate duality. Conjugate duality for an indicator function is equivalent to basic separation. To prove that a closed, convex set is the intersection of its containing, closed, half spaces, we calculate the support to the indicator of these half spaces. Applying this result proves conjugate duality. For a proper, closed, convex function, its epigraph is the intersection of all nonvertical containing, closed, half spaces. A calculation establishes epi f**=epi f. The containing half-space symmetry derived below furnishes a geometric interpretation of conjugate duality: the nonvertical containing half spaces of the epigraph define the epigraph of the conjugate.

Date: 2026
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DOI: 10.1007/978-3-032-21396-9_4

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