## Attouch-Wets convergence and a differential operator for convex functions

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- by Gerald Beer and Michel Théra PDF
- Proc. Amer. Math. Soc.
**122**(1994), 851-858 Request permission

## Abstract:

The purpose of this note is to characterize Attouch-Wets convergence for sequences of proper lower semicontinuous convex functions defined on a Banach space*X*in terms of the behavior of an operator $\Delta$ defined on the space of such functions with values in $X \times R \times {X^ \ast }$, defined by $\Delta (f) = \{ (x,f(x),y):(x,y) \in \partial f\}$. We show that $\langle {f_n}\rangle$ is Attouch-Wets convergent to

*f*if and only if points of $\Delta (f)$ lying in a fixed bounded set can be uniformly approximated by points of $\Delta ({f_n})$ for large

*n*. The operator $\Delta$ is a natural carrier of the Borwein variational principle, which is a key tool in both directions of our proof.

## References

- Hédy Attouch and Gerald Beer,
*On the convergence of subdifferentials of convex functions*, Arch. Math. (Basel)**60**(1993), no. 4, 389–400. MR**1206324**, DOI 10.1007/BF01207197 - Hédy Attouch, Roberto Lucchetti, and Roger J.-B. Wets,
*The topology of the $\rho$-Hausdorff distance*, Ann. Mat. Pura Appl. (4)**160**(1991), 303–320 (1992). MR**1163212**, DOI 10.1007/BF01764131 - H. Attouch, J. L. Ndoutoume, and M. Théra,
*Epigraphical convergence of functions and convergence of their derivatives in Banach spaces*, Sém. Anal. Convexe**20**(1990), Exp. No. 9, 45. MR**1114679** - Hédy Attouch and Roger J.-B. Wets,
*Isometries for the Legendre-Fenchel transform*, Trans. Amer. Math. Soc.**296**(1986), no. 1, 33–60. MR**837797**, DOI 10.1090/S0002-9947-1986-0837797-X - Hédy Attouch and Roger J.-B. Wets,
*Quantitative stability of variational systems. I. The epigraphical distance*, Trans. Amer. Math. Soc.**328**(1991), no. 2, 695–729. MR**1018570**, DOI 10.1090/S0002-9947-1991-1018570-0 - Dominique Azé,
*Caractérisation de la convergence au sens de Mosco en terme d’approximations inf-convolutives*, Ann. Fac. Sci. Toulouse Math. (5)**8**(1986/87), no. 3, 293–314 (French, with English summary). MR**948756** - D. Azé and J.-P. Penot,
*Operations on convergent families of sets and functions*, Optimization**21**(1990), no. 4, 521–534. MR**1069660**, DOI 10.1080/02331939008843576 - Gerald Beer,
*Conjugate convex functions and the epi-distance topology*, Proc. Amer. Math. Soc.**108**(1990), no. 1, 117–126. MR**982400**, DOI 10.1090/S0002-9939-1990-0982400-8 - Gerald Beer,
*The slice topology: a viable alternative to Mosco convergence in nonreflexive spaces*, Nonlinear Anal.**19**(1992), no. 3, 271–290. MR**1176063**, DOI 10.1016/0362-546X(92)90145-5 - Gerald Beer,
*Wijsman convergence of convex sets under renorming*, Nonlinear Anal.**22**(1994), no. 2, 207–216. MR**1258956**, DOI 10.1016/0362-546X(94)90034-5 - Gerald Beer,
*Lipschitz regularization and the convergence of convex functions*, Numer. Funct. Anal. Optim.**15**(1994), no. 1-2, 31–46. MR**1261596**, DOI 10.1080/01630569408816547 - Gerald Beer and Jonathan M. Borwein,
*Mosco and slice convergence of level sets and graphs of linear functionals*, J. Math. Anal. Appl.**175**(1993), no. 1, 53–67. MR**1216744**, DOI 10.1006/jmaa.1993.1151 - Gerald Beer and Roberto Lucchetti,
*Convex optimization and the epi-distance topology*, Trans. Amer. Math. Soc.**327**(1991), no. 2, 795–813. MR**1012526**, DOI 10.1090/S0002-9947-1991-1012526-X - J. M. Borwein,
*A note on $\varepsilon$-subgradients and maximal monotonicity*, Pacific J. Math.**103**(1982), no. 2, 307–314. MR**705231** - S. Fitzpatrick and R. R. Phelps,
*Bounded approximants to monotone operators on Banach spaces*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**9**(1992), no. 5, 573–595 (English, with English and French summaries). MR**1191009**, DOI 10.1016/S0294-1449(16)30230-X - J.-B. Hiriart-Urruty,
*Lipschitz $r$-continuity of the approximate subdifferential of a convex function*, Math. Scand.**47**(1980), no. 1, 123–134. MR**600082**, DOI 10.7146/math.scand.a-11878
—, - Umberto Mosco,
*Convergence of convex sets and of solutions of variational inequalities*, Advances in Math.**3**(1969), 510–585. MR**298508**, DOI 10.1016/0001-8708(69)90009-7 - Umberto Mosco,
*On the continuity of the Young-Fenchel transform*, J. Math. Anal. Appl.**35**(1971), 518–535. MR**283586**, DOI 10.1016/0022-247X(71)90200-9 - Robert R. Phelps,
*Convex functions, monotone operators and differentiability*, Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1989. MR**984602**, DOI 10.1007/BFb0089089 - R. Tyrrell Rockafellar,
*Convex analysis*, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR**0274683**

*Extensions of Lipschitz functions*, J. Math. Anal. Appl.

**77**(1980), 539-554.

## Additional Information

- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**122**(1994), 851-858 - MSC: Primary 49J45; Secondary 46N10, 49J50
- DOI: https://doi.org/10.1090/S0002-9939-1994-1204368-5
- MathSciNet review: 1204368