Fenchel duality, Fitzpatrick functions and the Kirszbraun-Valentine extension theorem
Authors:
Simeon Reich and Stephen Simons
Journal:
Proc. Amer. Math. Soc. 133 (2005), 2657-2660
MSC (2000):
Primary 46C05, 47H09; Secondary 46N10
DOI:
https://doi.org/10.1090/S0002-9939-05-07983-9
Published electronically:
March 22, 2005
MathSciNet review:
2146211
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We present a new proof of the classical Kirszbraun-Valentine extension theorem. Our proof is based on the Fenchel duality theorem from convex analysis and an analog for nonexpansive mappings of the Fitzpatrick function from monotone operator theory.
- 1. Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673
- 2. Haïm Brezis and Alain Haraux, Image d’une somme d’opérateurs monotones et applications, Israel J. Math. 23 (1976), no. 2, 165–186. MR 0399965, https://doi.org/10.1007/BF02756796
- 3. Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
- 4. Simon Fitzpatrick, Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 20, Austral. Nat. Univ., Canberra, 1988, pp. 59–65. MR 1009594
- 5. M. D. Kirszbraun, Über die zusammenziehende und Lipschitzsche Transformationen, Fund. Math. 22 (1934), 77-108.
- 6. Earl J. Mickle, On the extension of a transformation, Bull. Amer. Math. Soc. 55 (1949), 160–164. MR 0029974, https://doi.org/10.1090/S0002-9904-1949-09189-9
- 7. Simeon Reich, Extension problems for accretive sets in Banach spaces, J. Functional Analysis 26 (1977), no. 4, 378–395. MR 0477893
- 8. S. Reich, The range of sums of accretive and monotone operators, J. Math. Anal. Appl. 68 (1979), 310-317. MR 0531440 (80g:47060)
- 9. R. T. Rockafellar, Extension of Fenchel’s duality theorem for convex functions, Duke Math. J. 33 (1966), 81–89. MR 0187062
- 10. I. J. Schoenberg, On a theorem of Kirzbraun and Valentine, Amer. Math. Monthly 60 (1953), 620–622. MR 0058232, https://doi.org/10.2307/2307864
- 11. S. Simons and C. Zalinescu, Fenchel duality, Fitzpatrick functions and maximal monotonicity, J. Nonlinear Convex Anal., in press.
- 12. F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math. 67 (1945), 83–93. MR 0011702, https://doi.org/10.2307/2371917
- 13. J. H. Wells and L. R. Williams, Embeddings and extensions in analysis, Springer-Verlag, New York-Heidelberg, 1975. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. MR 0461107
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Additional Information
Simeon Reich
Affiliation:
Department of Mathematics, The Technion - Israel Institute of Technology, 32000 Haifa, Israel
Email:
sreich@tx.technion.ac.il
Stephen Simons
Affiliation:
Department of Mathematics, University of California, Santa Barbara, California 93106-3080
Email:
simons@math.ucsb.edu
DOI:
https://doi.org/10.1090/S0002-9939-05-07983-9
Received by editor(s):
April 21, 2004
Published electronically:
March 22, 2005
Communicated by:
Jonathan M. Borwein
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
