Epigraph

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

Chapter 6 in Conjugate Duality in Economic Analysis, 2026, pp 49-54 from Springer

Abstract: Abstract The epigraph is the points $$\left ( \boldsymbol {x},y\right )$$ above the function, $$\delta _{\mathrm {epi\ }f}\left ( \boldsymbol {x},y\right ) =\delta _{f\left ( \boldsymbol {x}\right ) \leq y}.$$ The conjugate is the support to the epigraph, $$f^{\ast }\left ( \boldsymbol {x}^{\ast }\right ) =\delta _{\mathrm {epi\ }f}^{\ast }\left ( \boldsymbol {x}^{\ast },-1\right )$$ . “Epi-multiplication” is the support to the epigraph of the conjugate: $$\left ( y\star f\right ) \left ( \boldsymbol {x}\right ) :=\delta _{\mathrm {epi\ }f^{\ast }}^{\ast }\left ( \boldsymbol {x},-y\right ) \!.$$ Here y is the “epi-multiplier.” That epi-multiplication corresponds to multiplying the epigraph by y is the source of the terminology. Epi-multiplication by zero is the “horizon function”: $$f^{\infty }\left ( \boldsymbol {x}\right ) :=\left ( 0\star f\right ) \left ( \boldsymbol {x}\right ) =\delta _{\mathrm {dom\ }f^{\ast }}^{\ast }\left ( \boldsymbol {x}\right ) \!,$$ the support to the effective domain of the conjugate.

Date: 2026
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DOI: 10.1007/978-3-032-21396-9_6

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