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The proximal average: Basic theory. (English) Zbl 1172.26003
The notion of proximal average of two convex functions introduced by H. H. Bauschke, E. Matoušková and S. Reich [Nonlinear Anal., Theory Methods Appl. 56A, No. 5, 715–738 (2004; Zbl 1059.47060)] is extended to finitely many convex functions on a real Hilbert space. The domain of the $$\lambda$$-weighted proximal average of $$f_{1},\dots,f_{n}$$, with $$\lambda =(\lambda_{1},\dots,\lambda _{n})$$, is shown to be $$\lambda _{1}\operatorname{dom} f+\dots+\lambda_{n} \operatorname{dom} f_{n}$$. It is also proved that the Fenchel conjugate of the $$\lambda$$-weighted proximal average of $$f_{1},\dots,f_{n}$$ with parameter $$\mu$$ is the $$\lambda$$-weighted proximal average of the convex conjugates $$f_{1}^{\ast },\dots,f_{n}^{\ast }$$ with parameter $$\mu ^{-1}.$$ From this result it follows that the proximal average is a lower semicontinuous proper convex function. The Moreau envelope, the proximal mapping and the subdifferential of the proximal average are also studied, and the arithmetical and epigraphical averages are shown to be pointwise limits of the proximal average. In the finite-dimensional case, the authors prove the epi-continuity of the proximal average.

##### MSC:
 26B25 Convexity of real functions of several variables, generalizations 26E60 Means
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