The cosmic Hausdorff topology, the bounded Hausdorff topology and continuity of polarity

Penot, Jean-Paul

doi:10.1090/S0002-9939-1991-1068129-X

The cosmic Hausdorff topology, the bounded Hausdorff topology and continuity of polarity
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Proc. Amer. Math. Soc. 113 (1991), 275-285 Request permission

Abstract:

We give geometric proofs of recent results of G. Beer [13]: the Young-Fenchel correspondence $f \to {f^*}$ is bicontinuous on the space of closed proper convex functions on a normed vector space $X$ endowed with the epidistance topology and the polarity operation is continuous on the space of closed convex subsets of $X$ with the bounded Hausdorff topology. Our methods are in the spirit of a famous result due to Walkup and Wets [31] about the isometric character of the polarity for closed convex cones. We also prove that the Hausdorff distance associated with the cosmic distance on the space of convex subsets of a normed vector space induces the bounded -Hausdorff topology. This shows a link between Beer’s results and the continuity results of [2].

References

H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 773850
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, Another isometry for the Legendre-Fenchel transform, J. Math. Anal. Appl. 131 (1988), no. 2, 404–411. MR 935277, DOI 10.1016/0022-247X(88)90214-4

Quantitative stability of variational systems

The epigraphical distance

Quantitative stability of variational systems

A framework for nonlinear conditioning

On some metric aspects of set convergence

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
Dominique Azé and Jean-Paul Penot, Recent quantitative results about the convergence of convex sets and functions, Functional analysis and approximation (Bagni di Lucca, 1988) Pitagora, Bologna, 1989, pp. 90–110. MR 1001572

Qualitative results about the convergence of convex sets and convex functions

Gerald Beer, On Mosco convergence of convex sets, Bull. Austral. Math. Soc. 38 (1988), no. 2, 239–253. MR 969914, DOI 10.1017/S0004972700027519
Gerald Beer, On the Young-Fenchel transform for convex functions, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1115–1123. MR 937844, DOI 10.1090/S0002-9939-1988-0937844-8
Gerald Beer, Convergence of continuous linear functionals and their level sets, Arch. Math. (Basel) 52 (1989), no. 5, 482–491. MR 998621, DOI 10.1007/BF01198356
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

Continuity results for the epi-distance topology with applications to constrained convex optimization problems

C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310

On the continuity of polar point-to-sets maps

Continuity of the Fenchel correspondence and continuity of polarity

Coercivité des intégrandes convexes normales. Application à la minimisation des fonctionnelles intégrales et du calcul des variations

J.-B. Hiriart-Urruty, A general formula on the conjugate of the difference of functions, Canad. Math. Bull. 29 (1986), no. 4, 482–485. MR 860858, DOI 10.4153/CMB-1986-076-7
Lars Hörmander, Sur la fonction d’appui des ensembles convexes dans un espace localement convexe, Ark. Mat. 3 (1955), 181–186 (French). MR 68112, DOI 10.1007/BF02589354
J.-L. Joly, Une famille de topologies sur l’ensemble des fonctions convexes pour lesquelles la polarité est bicontinue, J. Math. Pures Appl. (9) 52 (1973), 421–441 (1974) (French). MR 500129
P. Huard, Une approche synthétique des recherches linéaires dans les méthodes de pente, Bull. Direction Études Rech. Sér. C Math. Inform. 2 (1981), 3, 89–131 (1982) (French, with English summary). MR 658180
David G. Luenberger, Optimization by vector space methods, John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238472
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
Jean-Paul Penot, On regularity conditions in mathematical programming, Math. Programming Stud. 19 (1982), 167–199. Optimality and stability in mathematical programming. MR 669731, DOI 10.1007/bfb0120988

Preservation of persistence and stability under intersections and operations

Topologies and convergences on the set of convex functions

Convex Analysis

Sur une forme de continuité du passage au cône polaire

David W. Walkup and Roger J.-B. Wets, Continuity of some convex-cone-valued mappings, Proc. Amer. Math. Soc. 18 (1967), 229–235. MR 209806, DOI 10.1090/S0002-9939-1967-0209806-6
R. J.-B. Wets, Convergence of convex functions, variational inequalities and convex optimization problems, Variational inequalities and complementarity problems (Proc. Internat. School, Erice, 1978) Wiley, Chichester, 1980, pp. 375–403. MR 578760