On supportless convex sets

Borwein, J. M.; Tingley, D. W.

doi:10.1090/S0002-9939-1985-0787897-2

On supportless convex sets
HTML articles powered by AMS MathViewer

by J. M. Borwein and D. W. Tingley PDF

Proc. Amer. Math. Soc. 94 (1985), 471-476 Request permission

Abstract:

We give some general constructions of supportless convex subsets of normed spaces and pose a number of open questions.

References

Errett Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961), 97–98. MR 123174, DOI 10.1090/S0002-9904-1961-10514-4
Errett Bishop and R. R. Phelps, The support functionals of a convex set, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 27–35. MR 0154092
J. Borwein and R. O’Brien, Tangent cones and convexity, Canad. Math. Bull. 19 (1976), no. 3, 257–261. MR 454586, DOI 10.4153/CMB-1976-040-x

Convex proximal sets with nonsurjective projections

Topological vector spaces

A. Brøndsted and R. T. Rockafellar, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965), 605–611. MR 178103, DOI 10.1090/S0002-9939-1965-0178103-8
Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094
I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. MR 346619, DOI 10.1016/0022-247X(74)90025-0
Ivar Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 3, 443–474. MR 526967, DOI 10.1090/S0273-0979-1979-14595-6
Ivar Ekeland and Gérard Lebourg, Generic Fréchet-differentiability and perturbed optimization problems in Banach spaces, Trans. Amer. Math. Soc. 224 (1976), no. 2, 193–216 (1977). MR 431253, DOI 10.1090/S0002-9947-1976-0431253-2
P. A. Fillmore and J. P. Williams, On operator ranges, Advances in Math. 7 (1971), 254–281. MR 293441, DOI 10.1016/S0001-8708(71)80006-3
Seymour Goldberg and A. H. Kruse, The existence of compact linear maps between Banach spaces, Proc. Amer. Math. Soc. 13 (1962), 808–811. MR 141971, DOI 10.1090/S0002-9939-1962-0141971-7
Richard B. Holmes, Geometric functional analysis and its applications, Graduate Texts in Mathematics, No. 24, Springer-Verlag, New York-Heidelberg, 1975. MR 0410335
V. L. Klee Jr., Extremal structure of convex sets. II, Math. Z. 69 (1958), 90–104. MR 92113, DOI 10.1007/BF01187394
Victor Klee, On a question of Bishop and Phelps, Amer. J. Math. 85 (1963), 95–98. MR 149241, DOI 10.2307/2373188

Research problems

72

Stephen Saxon and Mark Levin, Every countable-codimensional subspace of a barrelled space is barrelled, Proc. Amer. Math. Soc. 29 (1971), 91–96. MR 280972, DOI 10.1090/S0002-9939-1971-0280972-0
N. T. Peck, Support points in locally convex spaces, Duke Math. J. 38 (1971), 271–278. MR 282191
J. Arias de Reyna, Dense hyperplanes of first category, Math. Ann. 249 (1980), no. 2, 111–114. MR 578718, DOI 10.1007/BF01351408
Francis Sullivan, A characterization of complete metric spaces, Proc. Amer. Math. Soc. 83 (1981), no. 2, 345–346. MR 624927, DOI 10.1090/S0002-9939-1981-0624927-9
F. Sullivan, Ordering and completeness of metric spaces, Nieuw Arch. Wisk. (3) 29 (1981), no. 2, 178–193. MR 641675
Aaron R. Todd and Stephen A. Saxon, A property of locally convex Baire spaces, Math. Ann. 206 (1973), 23–34. MR 328523, DOI 10.1007/BF01431526
Jay S. Treiman, Characterization of Clarke’s tangent and normal cones in finite and infinite dimensions, Nonlinear Anal. 7 (1983), no. 7, 771–783. MR 707085, DOI 10.1016/0362-546X(83)90033-0
Hoàng Tụy, A fixed point theorem involving a hybrid inwardness-contraction condition, Math. Nachr. 102 (1981), 271–275. MR 642157, DOI 10.1002/mana.19811020122