Journal of Convex Analysis 32 (2025), No. 3, 877--882
Copyright Heldermann Verlag 2025
A Convex, Finite and Lower Semicontinuous Function with Empty Subdifferential
Gerd Wachsmuth
Institute of Mathematics, Brandenburgische Technische Universität Cottbus-Senftenberg, Cottbus, Germany
wachsmuth@b-tu.de
We give an example of a convex, finite and lower semicontinuous function whose subdifferential is everywhere empty. This is possible since the function is defined on an incomplete normed space. The function serves as a universal counterexample to various statements in convex analysis in which completeness is required.
Keywords: Subdifferential, incomplete space, Fenchel duality, convex sum rule.
MSC: 46N10, 90C25.
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