Abstract

We give a systematic and thorough study of geometric notions and results connected to Minkowski's measure of symmetry and the extension of the well-known Minkowski functional to arbitrary, not necessarily symmetric convex bodies K on any (real) normed space X. Although many of the notions and results we treat in this paper can be found elsewhere in the literature, they are scattered and possibly hard to find. Further, we are not aware of a systematic study of this kind and we feel that several features, connections and properties – e.g. the connections between many equivalent formulations – are new, more general and they are put in a better perspective now. In particular, we prove a number of fundamental properties of the extended Minkowski functional α(K,x)\alpha(K,\mathbf{x}), including convexity, global Lipschitz boundedness, linear growth and approximation of the classical Minkowski functional of the central symmetrization of the body K. Our aim is to present how in the recent years these notions proved to be surprisingly relevant and effective in problems of approximation theory.

Author information

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

Szilárd Gy. Révész

A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O.Box 127, 1364 Budapest,
Hungary

revesz@renyi.hu

Yannis Sarantopoulos

National Technical University, School of Applied Mathematical and Physical Sciences,
Dept. of Mathematics, Zografou Campus, 15780 Athens, Greece

ysarant@math.ntua.gr

Keywords

  • convex body
  • support function
  • supporting hyperplanes
  • halfspaces and layers
  • Minkowski functional
  • convex functions in normed spaces
  • Lipschitz bounds
  • central symmetrization
  • centroid
  • cone of convex bodies
  • measure of symmetry
  • width of K in a direction
  • homothetic transformations
  • existence of minima of continuous convex functions in normed spaces
  • separation of convex sets
  • multivariate polynomials
  • Bernstein and Chebyshev type extremal problems for multivariate polynomials

Mathematics Subject Classification

Primary 46B20; 41A17, 41A63, 41A44, 26D05

References

  1. A. V. Andrianov: On some open problems for algebraic polynomials on bounded convex bodies, East J. Approx. 5(1) (1999) 117–123.Google Scholar
  2. M. Baran: Bernstein type theorems for compact sets in Rn revisited, J. Approx. Theory 79 (1994) 190–198.https://doi.org/10.1006/jath.1994.1124Google Scholar
  3. I. Bárány: The technique of M -regions and cap coverings: A survey, Rend. Circ. Mat. Palermo (2) Suppl. 65 (2000) 21–38.Google Scholar
  4. I. Bárány, D. G. Larman: Convex bodies, economic cap coverings, random polytopes, Mathematika 35 (1988) 274–291.https://doi.org/10.1112/S0025579300015266Google Scholar
  5. S. Dineen: Complex Analysis on Infinite Dimensional Spaces, Springer Monographs in Mathematics, Springer (1999).https://doi.org/10.1007/978-1-4471-0869-6Google Scholar
  6. R. E. Edwards: Functional Analysis, Holt Rinehart & Winston, New York et al. (1965).Google Scholar
  7. B. Grünbaum: Measures of symmetry for convex sets, Proc. Sympos. Pure Math. 7 (1963) 233–270.https://doi.org/10.1090/pspum/007/0156259Google Scholar
  8. P. C. Hammer: Covex bodies, associated with a convex body, Proc. Amer. Math. Soc. 2 (1951) 781–793.https://doi.org/10.1090/S0002-9939-1951-0052802-0Google Scholar
  9. P. C. Hammer, A. Sobczyk: Critical points of a convex body, Abstract 112, Bull. Amer. Math. Soc. 2 (1951) 127.https://doi.org/10.1090/S0002-9939-1951-0052802-0Google Scholar
  10. Handbook of Convex Geometry. Volume A, P. M. Gruber, J. M. Wills (eds.), North-Holland, Amsterdam (1993).Google Scholar
  11. L. Hörmander: Notions of Convexity, Progress in Mathematics 127, Birkhäuser, Basel (1994).Google Scholar
  12. L. Hörmander: Sur la fonction d’appui des ensembles convexes dans un espace localement convexe, Arkiv för Mat. 3 (1955) 181–186.https://doi.org/10.1007/BF02589354Google Scholar
  13. P. Huard: Resolution of mathematical programming with nonlinear constraints by the method of centers, in: Nonlinear Programming, J. Aladie (ed.), North-Holland (1967) 209–219.Google Scholar
  14. V. L. Klee: The critical set of a convex body, Amer. J. Math. 75 (1953) 178–188.https://doi.org/10.2307/2372627Google Scholar
  15. A. Kroó: Universal polynomial majorants on convex bodies, J. Approx. Theory 111 (2001) 220–232.https://doi.org/10.1006/jath.2001.3573Google Scholar
  16. A. Kroó, Sz. Gy. Révész: On Bernstein and Markov-type inequalities for multivariate polynomials on convex bodies, J. Approx. Theory 99 (1999) 134–152.https://doi.org/10.1006/jath.1998.3314Google Scholar
  17. A. Kroó, D. Schmidt: Some extremal problems for multivariate polynomials on convex bodies, J. Approx. Theory 90(3) (1997) 415–434.https://doi.org/10.1006/jath.1996.3083Google Scholar
  18. E. Makai: Personal communication, July 2002.Google Scholar
  19. G. V. Milovanović, D. S. Mitrinović, Th. M. Rassias: Topics in Polynomials: Extremal Problems, Inequalities, Zero, World Scientific, Singapore (1994).https://doi.org/10.1142/1284Google Scholar
  20. H. Minkowski: Allgemeine Lehrsätze über die konvexen Polyeder, Gött. Nachr. (1897) 198–219; Ges. Abh. 2 (1911) 103–121, Leipzig, Berlin.Google Scholar
  21. B. H. Neumann: On some affine invariants of closed convex regions, J. London Math. Soc. 14 (1939) 262–272.https://doi.org/10.1112/jlms/s1-14.4.262Google Scholar
  22. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics 1364, 2nd Ed., Springer, Berlin (1993).Google Scholar
  23. J. Radon: Über eine Erweiterung des Begriffs der konvexen Funktionen, mit einer Anwendung auf die Theorie der konvexen Körper, S.-B. Akad Wiss. Wien 125 (1916) 241–258.Google Scholar
  24. Sz. Gy. Révész: Uniqueness of multivariate Chebyshev-type extremal polynomials for convex bodies, East J. Approx. 7 (2001) 205–240.Google Scholar
  25. Sz. Gy. Révész, Y. Sarantopoulos: Chebyshev’s extremal problems of polynomial growth in real normed spaces, J. Contemporary Analysis Applications, 36(5) (2001) 62–81.Google Scholar
  26. T. J. Rivlin: The Chebyshev Polynomials, Wiley, New York (1974).Google Scholar
  27. T. J. Rivlin, H. S. Shapiro: A unified approach to certain problems of approximation and minimization, J. Soc. Ind. Appl. Math. 9 (1961) 670–699.https://doi.org/10.1137/0109056Google Scholar
  28. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  29. W. Rudin: Functional Analysis, 2nd Ed., McGraw-Hill, New York (1991).Google Scholar
  30. Y. Sarantopoulos: Bounds on the derivatives of polynomials on Banach spaces, Math. Proc. Cambridge Philos. Soc. 110 (1991) 307–312.https://doi.org/10.1017/S0305004100070389Google Scholar
  31. R. Schneider: Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia in Mathematics and Applications 44, Cambridge University Press (1993).https://doi.org/10.1017/CBO9780511526282Google Scholar
  32. R. Schneider: Personal communication, August 2001.Google Scholar
  33. V. I. Skalyga: Analogues of the Markov and Bernstein inequalities on convex bodies in Banach spaces, Izvestiya: Mathematics 62(2) (1998) 375–397.https://doi.org/10.1070/IM1998v062n02ABEH000179Google Scholar
  34. V. I. Skalyga: Analogs of the Markov and Schaeffer–Duffin Inequalities for Convex Bodies, Mathematical Notes 68(1) (2000) 130–134.https://doi.org/10.1007/BF02674657Google Scholar
  35. V. I. Skalyga: Estimates of the Derivatives of Polynomials on Convex Bodies, Proc. Steklov Inst. Math. 218 (1997) 372–383.Google Scholar
  36. R. Webster: Convexity, Oxford University Press, Oxford (1994).https://doi.org/10.1093/oso/9780198531470.001.0001Google Scholar
  37. E. Werner: Illumination bodies and the affine surface area, Studia Math. 110 (1994) 257–269.https://doi.org/10.4064/sm-110-3-257-269Google Scholar

Suggested citation

S. G. Révész, Y. Sarantopoulos. “The Generalized Minkowski Functional with Applications in Approximation Theory.” Journal of Convex Analysis 11 (2004), No. 2, 303–334. https://doi.org/10.68381/jca11020

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2004. Rights now held by Banach Press.