Integral-Type Representations for the Subdifferential of Suprema

Abstract An integral-like representation is provided for the $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> -subdifferential of the supremum of an arbitrary family of convex functions. Our characterizations are expressed through appropriate discrete sums performed on the data functions $$f_{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> ’s together with specific singular measures operating on them. Moreover, as long as the underlying space is a reflexive Banach space or a separable normed space, we substitute these additional measures with related limits that involve the data functions. All the objects involved in our characterizations rely intrinsically on the data functions that are (almost) active at the reference point.