Abstract

We investigate metric projections and distance functions referring to convex bodies in finite-dimensional normed spaces. For this purpose we identify the vector space with its dual space by using, instead of the usual identification via the standard inner product, the Legendre transform associated with the given norm. This approach yields re-interpretations of various properties of convex functions, and new relations between such functions and geometric properties of the studied norm are also derived. Here the concept of Birkhoff orthogonality, mostly common in finite-dimensional normed spaces, will play an important role.

Author information

Contact details are reproduced from the original publication and may be historical.

Vitor Balestro

Inst. de Matemática e Estatística, Universidade Federal Fluminense, 24210201 Niterói, Brazil

vitorbalestro@id.uff.br

Ralph Teixeira

Inst. de Matemática e Estatística, Universidade Federal Fluminense, 24210201 Niterói, Brazil

ralphct@id.uff.br

Keywords

  • Birkhoff orthogonality
  • convex body
  • convex functions
  • (differentiability of) distance functions
  • Legendre transform
  • subgradient

Mathematics Subject Classification

41A50, 41A65, 46B20, 46G05, 52A20, 52A21, 53C23, 58C20

References

  1. A. R. Alimov, I. G. Tsar’kov: Connectedness and other geometric properties of suns and Chebyshev sets (Russian), Fundam. Prikl. Mat. 19/4 (2014) 21–91; English translation in J. Math. Sci. (N.Y.) 217/6 (2016) 683–730.https://doi.org/10.1007/s10958-016-3000-1Google Scholar
  2. J. Alonso, H. Martini, S. Wu: On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequationes Math. 83/1-2 (2012) 153–189.https://doi.org/10.1007/s00010-011-0092-zGoogle Scholar
  3. J. C. Álvarez-Paiva, A. C. Thompson: Volumes in normed and Finsler spaces, in: A Sampler of Riemann-Finsler Geometry, D. Bao, R. Bryant, S. S. Chern, Z. Shen (eds.), Cambridge University Press, Cambridge (2004) 1–49.https://doi.org/10.1017/9781009701280.002Google Scholar
  4. E. Asplund: Differentiability of the metric projection in finite-dimensional Euclidean space, Proc. Amer. Math. Soc. 38 (1973) 218–219.https://doi.org/10.1090/S0002-9939-1973-0310150-XGoogle Scholar
  5. J. M. Borwein, S. P. Fitzpatrick, J. R. Giles: The differentiability of real functions on normed linear space using generalized subgradients, J. Math. Analysis Appl. 128/2 (1987) 512–534.https://doi.org/10.1016/0022-247X(87)90203-4Google Scholar
  6. J. M. Borwein, A. S. Lewis: Convex Analysis and Nonlinear Optimization: Theory and Examples, 2nd edition, CMS Books in Mathematics 3, Canadian Mathematical Society, Ottawa (2006).https://doi.org/10.1007/978-0-387-31256-9Google Scholar
  7. J. M. Borwein, J. D. Vanderwerff: Convex Functions: Constructions, Characterizations and Counterexamples, Encyclopedia of Mathematics and Its Applications 109, Cambridge University Press, Cambridge (2010).https://doi.org/10.1017/CBO9781139087322Google Scholar
  8. A. L. Brown: Suns in normed linear spaces which are finite-dimensional, Math. Annalen 279/1 (1987) 87–101.https://doi.org/10.1007/BF01456192Google Scholar
  9. L. Bunt: Bijdrage tot de theorie der konvekse puntverzamelingen, Thesis, University of Groningen, Amsterdam (1934).Google Scholar
  10. F. H. Clarke: Optimization and Nonsmooth Analysis, Classics in Applied Mathematics 5, Society for Industrial and Applied Mathematics, Philadelphia (1990).https://doi.org/10.1137/1.9781611971309Google Scholar
  11. R. Correa, D. Salas, L. Thibault: Smoothness of the metric projection onto nonconvex bodies in Hilbert spaces, J. Math. Analysis Appl. 457/2 (2018) 1307–1332.https://doi.org/10.1016/j.jmaa.2016.08.064Google Scholar
  12. F. Deutsch: The convexity of Chebyshev sets in Hilbert space, in: Topics in Polynomials of One and Several Variables and Applications, World Scientific Publishing, New York (1993) 143–150.https://doi.org/10.1142/9789814360296_0011Google Scholar
  13. S. Fitzpatrick: Metric projections and the differentiability of distance functions, Bull. Aust. Math. Soc. 22 (1980) 291–312.https://doi.org/10.1017/S0004972700006596Google Scholar
  14. J. R. Giles: Differentiability of distance functions and a proximinal property inducing convexity, Proc. Amer. Math. Soc. 104/2 (1988) 458–464.https://doi.org/10.1090/S0002-9939-1988-0962813-1Google Scholar
  15. J. R. Giles: A distance function property implying differentiability, Bull. Aust. Math. Soc. 39/1 (1989) 59–70.https://doi.org/10.1017/S0004972700027982Google Scholar
  16. L. Hetzelt: On suns and cosuns in finite-dimensional normed real vector spaces, Acta Math. Hungar. 45/1-2 (1985) 53–68.https://doi.org/10.1007/BF01955023Google Scholar
  17. Á. G. Horváth, Z. Lángi, M. Spirova: Semi-inner products and the concept of semi-polarity, Results Math. 71/1-2 (2017) 127–144.https://doi.org/10.1007/s00025-015-0510-yGoogle Scholar
  18. L. A. Karlovitz: The construction and application of contractive retractions in 2-dimensional normed linear spaces, Indiana Univ. Math. J. 22/5 (1972) 473–481.https://doi.org/10.1512/iumj.1973.22.22041Google Scholar
  19. S. G. Krantz, H. R. Parks: On the vector sum of two convex sets in space, Canad. J. Math. 43/2 (1991) 347–355.https://doi.org/10.4153/CJM-1991-020-7Google Scholar
  20. Y. Li, L. Nirenberg: Regularity of the distance function to the boundary, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 29/1 (2005) 257–264.Google Scholar
  21. H. Martini, K. J. Swanepoel, G. Weiss,: The geometry of Minkowski spaces – a survey. Part I, Expositiones Math. 19 (2001) 97–142.https://doi.org/10.1016/S0723-0869(01)80025-6Google Scholar
  22. D. Noll: Directional differentiability of the metric projection in Hilbert space, Pacific J. Math. 170/2 (1995) 567–592.https://doi.org/10.2140/pjm.1995.170.567Google Scholar
  23. J. P. Penot, R. Ratsimahalo: Characterizations of metric projections in Banach spaces and applications, Abstr. Appl. Analysis 3/1-2 (1998) 85–103.https://doi.org/10.1155/S1085337598000451Google Scholar
  24. R. A. Poliquin, R. T. Rockafellar, L. Thibault: Local differentiability of distance functions, Trans. Amer. Math. Soc. 352/11 (2000) 5231–5249.https://doi.org/10.1090/S0002-9947-00-02550-2Google Scholar
  25. R. T. Rockafellar: Convex Analysis, Princeton Landmarks in Mathematics and Physics, Princeton University Press, Princeton (1970).Google Scholar
  26. M. Safdari: The distance function from the boundary of a domain with corners, Nonlinear Analysis 181 (2019) 294–310.https://doi.org/10.1016/j.na.2019.01.004Google Scholar
  27. D. Sain, K. Paul, A. Mal: On approximate Birkhoff-James orthogonality and normal cones in a normed space, J. Convex Analysis 26/1 (2019) 341–351.https://doi.org/10.68381/jca26019Google Scholar
  28. Sangeeta, T. D. Narang: On suns in linear metric spaces and convex metric spaces, East J. Approx. 17/2 (2011) 127–139.Google Scholar
  29. R. Schneider: Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications 151, Cambridge University Press, Cambridge (2014).Google Scholar
  30. A. C. Thompson: Minkowski Geometry, Encyclopedia of Mathematics and Its Applications 63, Cambridge University Press, Cambridge (1996).Google Scholar
  31. L. P. Vlasov: On almost convex sets in Banach spaces (Russian), Dokl. Akad. Nauk SSSR 163 (1965) 18–21.Google Scholar
  32. L. P. Vlasov: Approximatively convex sets in uniformly smooth spaces, Mat. Zametki 1 (1967) 443–450.https://doi.org/10.1007/BF01095547Google Scholar
  33. L. Zajíček: Differentiability of the distance function and points of multivaluedness of the metric projection in Banach space, Czechoslovak Math. J. 33/2 (1983) 292–308.https://doi.org/10.21136/CMJ.1983.101878Google Scholar
  34. L. Zajíček: On the Fréchet differentiability of distance functions, in: Proceedings of the 12th Winter School on Abstract Analysis, Srní 1984, Rend. Circ. Mat. Palermo 2, Suppl. No. 5 (1984) 161–165.Google Scholar
  35. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific Publishing, River Edge (2002).https://doi.org/10.1142/5021Google Scholar
  36. Z. Wu: A Chebyshev set and its distance function, J. Approximation Theory 119/2 (2002) 181–192.https://doi.org/10.1006/jath.2002.3728Google Scholar

Suggested citation

V. Balestro, H. Martini, R. Teixeira. “Convex Analysis in Normed Spaces and Metric Projections onto Convex Bodies.” Journal of Convex Analysis 28 (2021), No. 4, 1223–1248. https://doi.org/10.68381/jca28072

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2021. Rights now held by Banach Press.