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Convergence of convex functions and generalized inf-convolutive approximations. (Convergences des fonctions convexes et approximations inf-convolutives généralisées.) (French) Zbl 1086.49012

The authors prove that the slice convergence of a sequence \((f^n)_n\) of appropriate convex functions on a normed linear space \(X\) is equivalent to the slice convergence of its sequence of inf-convolution approximates of sufficiently small parameters associated to a kernel \(\Phi:X\to\mathbb R^+\), where the inf-convolution approximation of a convex \(f\) of parameter \(\lambda\) is \(f_\lambda=\inf\{f(u)+\Phi((x-u)/\lambda)\}\). It is equivalent with the pointwise convergence of certain regularized sequences. It is also shown that the Attouch–Wets convergence of \((f^n)_n\) is equivalent to the same type of convergence of its approximate sequences when \(\lambda\) converges to \(0\) and this is equivalent to the uniform convergence on bounded subsets of \(X\). With this, the authors generalize cases known earlier for \(\Phi =\| \cdot\| \).

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
26B25 Convexity of real functions of several variables, generalizations
52A41 Convex functions and convex programs in convex geometry
54B20 Hyperspaces in general topology
40A30 Convergence and divergence of series and sequences of functions
46B20 Geometry and structure of normed linear spaces
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