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On the level sum of two convex functions on Banach spaces. (English) Zbl 0876.49011

The authors are concerned with the semicontinuity, subdifferentiability and exactness of the level sum \(h{{+}\atop{t}}k\) of two convex functions \(h\) and \(k\) defined by \((h{{+}\atop{t}}k)(z)=\inf \{h(x)\vee k(y)\mid x+y=z\}\). In order to obtain such results they use the fact that \(\inf_{x\in X}F(x,0)=\max_{z^* \in Z^*}(-F^* (0,z^*))\) if \(X,Z\) are Banach spaces, \(F\) is a l.s.c.convex function and cone \(P_Z(\text{ dom} F)\) is a closed linear subspace, a result stated in Theorem 6 of reviewer’s paper [Z. Oper. Res., Ser A 31, A79–A101 (1987; Zbl 0695.49009)] and Proposition 3.1 of V. Jeyakumar [Nonlinear Anal., Theory Methods Appl. 15, No. 12, 1111-1122 (1990; Zbl 0724.90051)]. Similar results are stated in dual spaces and some examples are also given.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J52 Nonsmooth analysis
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