Cone Representation of a Convex Set

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

Chapter 19 in Conjugate Duality in Economic Analysis, 2026, pp 141-148 from Springer

Abstract: Abstract It is possible to replace the analysis of a closed, convex set by an equivalent analysis of a closed, convex cone. That a closed, convex set is the intersection of its containing closed half spaces $$\mathrm {epi\ }\delta _{C}^{\ast }$$ is the key: the cone representation of the set is the polar of the epigraph of the support. We derive various theoretical results about the cone representation via algebraic calculation. For a closed, convex set C, $$\mathrm {cone\ }C=\left \{\left ( \boldsymbol {x},-y\right ) \left | \begin {array}[c]{l@{}}\boldsymbol {x}\in yC{\text { if }}y>0\\ \boldsymbol {x}\in C^{\infty }{\text { if }}y=0 \end {array} \right . \right \}\!.$$ The barrier cone is the effective domain of the support, the directions in which the set is bounded. The horizon cone C∞ is the polar of the barrier cone and describes the set “at infinity,” the directions in which the set is unbounded. The concept of the polar of a cone generalizes to the polar of a set. The gauge is the support to the polar.

Date: 2026
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DOI: 10.1007/978-3-032-21396-9_19

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