Cone Representation of a Convex Function

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

Chapter 20 in Conjugate Duality in Economic Analysis, 2026, pp 149-156 from Springer

Abstract: Abstract Invoking the cone representation of a set, we define the cone representation of a convex function. We work out its properties via calculation and perturbation duality. The cone representation of a convex function f is the cone representation of its epigraph:$$\mathrm {cone}\,f:=\mathrm {cone}\left ( \mathrm {epi\ }f\right ) \!.$$ We calculate$$\mathrm {cone\ }f=\left \{\left ( \boldsymbol {x},y,-z\right ) \left | \begin {array}[c]{l@{}} \left ( \boldsymbol {x},y\right ) \in z\,\mathrm {epi\ }f{\text { if }}z>0\\ \left ( \boldsymbol {x},y\right ) \in \left ( \mathrm {epi\ }f\right ) ^{\infty }=\mathrm {epi\ }f^{\infty }{\text { if }}z=0. \end {array} \right . \right \}$$ The cone representation is the polar of the cone of half spaces containing the epigraph. The cone representation cone f* is almost the polar of cone f, but the final two arguments are in reverse order. Under general conditions, the polar of the cone generated by a contour set is the cone generated by the contour set for the conjugate. We interpret the Fenchel equality for the cone representation, for the standard case, the exceptional case, and the double exceptional case.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:conchp:978-3-032-21396-9_20

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DOI: 10.1007/978-3-032-21396-9_20

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