Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

PLANK PROBLEMS, POLARIZATION AND CHEBYSHEV CONSTANTS

  • Revesz, Szilard-Gy. (A. Renyi Institute of Mathematics Hungarian Academy of Sciences Budapest) ;
  • Sarantopoulos, Yannis (National Technical University School of Applied Mathematical and Physical Sciences Department of Mathematics)
  • Published : 2004.01.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this work we discuss "plank problems" for complex Banach spaces and in particular for the classical $L^{p}(\mu)$ spaces. In the case $1\;{\leq}\;p\;{\leq}\;2$ we obtain optimal results and for finite dimensional complex Banach spaces, in a special case, we have improved an early result by K. Ball [3]. By using these results, in some cases we are able to find best possible lower bounds for the norms of homogeneous polynomials which are products of linear forms. In particular, we give an estimate in the case of a real Hilbert space which seems to be a difficult problem. We have also obtained some results on the so-called n-th (linear) polarization constant of a Banach space which is an isometric property of the space. Finally, known polynomial inequalities have been derived as simple consequences of various results related to plank problems.

Keywords

References

  1. Polarization constants for products of linear functionals over $\mathbb{R}^2$ and $\mathbb{C}^2$ and Chebyshev constants of the unit sphere V.Anagnostopoulos;Sz.Revesz
  2. Linear Algebra Appl. v.285 Gaussian variables, polynomials and permanents J.Arias-de-Reyna https://doi.org/10.1016/S0024-3795(98)10125-8
  3. Invent. Math. v.104 The plank problem for symmetric bodies K.M.Ball https://doi.org/10.1007/BF01245089
  4. Bull. London Math. Soc. v.33 The complex plank problem K.M.Ball https://doi.org/10.1112/S002460930100813X
  5. Proc. Amer. Math. Soc. v.2 A solution of the plank problem T.Bang https://doi.org/10.2307/2031721
  6. Math. Proc. Cambridge Philos. Soc. v.124 Lower bounds for norms of products of polynomials C.Benitez;Y.Sarantopoulos;A.M.Tonge https://doi.org/10.1017/S030500419800259X
  7. Springer Monographs in Mathematics Complex analysis on infinite dimensional spaces S.Dineen
  8. Math. Z. v.17 Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzahliqen Koeffizienten M.Fekete https://doi.org/10.1007/BF01504345
  9. Summing and nuclear norms in Banach space theory G.J.O.Jameson
  10. Courant Aniversary Volume Extremum problems with inequalities as subsidiary conditions F.John
  11. Private communication V.M.Kadets
  12. Bull. London Math. Soc. v.31 A sharp version of Mahler's inequality for products of polynomials A.Kroo;I.E.Pritsker https://doi.org/10.1112/S0024609398005670
  13. Studia Math. v.63 Finite dimensional subspaces of ${L}_{p}$ D.Lewis https://doi.org/10.4064/sm-63-2-207-212
  14. J. Math. Mech. v.16 Proofs of some conjectures on permanents E.H.Lieb
  15. Inequalities: selecta of Elliot H. Lieb. E.H.Lieb
  16. The university of Texas at Austin, Txas Functional Analysis Seminar Projections of minimal norm, Longhorn Notes G.G.Lorentz;N. Tomczak- Jaegermann
  17. Mathematika v.7 An application of Jensen's formula to polynomials K.Mahler https://doi.org/10.1112/S0025579300001637
  18. Bull. Amer. Math. Soc. v.69 The permanent analogue of the Hadamard determinant theorem M.Marcus https://doi.org/10.1090/S0002-9904-1963-10975-1
  19. Proc. Amer. Math. Soc. v.15 The Hadamard theorem for permanents M.Marcus https://doi.org/10.2307/2034919
  20. Private communication M.Marcus
  21. Trans. Amer. Math. Soc. v.104 The Pythagorean theorem in certain symmetry classes of tensors M.Marcus;H.Minc https://doi.org/10.2307/1993799
  22. Amer. Math. Monthly v.72 Permanents M.Marcus;H.Minc https://doi.org/10.2307/2313846
  23. Funktsional Anal. i Prilozhen Appl. v.5 A new proof of the theorem of A. Dvoretzky on sections of convex bodies V.D.Milman
  24. Engl. transl.: Funct Anal. Appl. v.5 A new proof of the theorem of A. Dvoretzky on sections of convex bodies V.D.Milman
  25. Studia Math. v.134 Complexifications of real Banach spaces, polynomials and multilinear maps G.Munoz;Y.Sarantopoulos;A.M.Tonge
  26. The volume of convex bodies and Banach space geometry G.Pisier
  27. Problems and Theorems in Analysis I(Reprint of the 1st ed) G.Polya;G.Szego
  28. Math. Proc. Cambridge Philos. Soc. v.99 Estimates for polynomial norms on ${L}^{p}({\mu})$ spaces Y.Sarantopoulos https://doi.org/10.1017/S0305004100064185
  29. Ph.D. thesis. Brunel University Polynomials and multilinear mappings in Banach spaces Y.Sarantopoulos https://doi.org/10.1017/S0305004100064185
  30. J. Math. Anal. Appl. v.221 Geometry of spaces of polynomials R.A.Ryan;B.Turett https://doi.org/10.1006/jmaa.1998.5942
  31. Tohoku Math. J. v.44 Additions to the theory of polynomials in normed linear spaces A.E.Taylor
  32. Private communication A.M.Tonge
  33. J. Australian Math. Soc. Ser. v.A47 On the product of distances to a point set on a sphere G.Wagner
  34. Cambridge stud. Adv. Math. v.25 Banach spaces for analysts P.Wojtaszczyk

Cited by

  1. Numerical Plank Problem vol.50, pp.2, 2010, https://doi.org/10.5666/KMJ.2010.50.2.289
  2. Plank type problems for polynomials on Banach spaces vol.396, pp.2, 2012, https://doi.org/10.1016/j.jmaa.2012.06.029
  3. The dth linear polarization constant of Rd vol.255, pp.10, 2008, https://doi.org/10.1016/j.jfa.2008.06.011
  4. Potential Theoretic Approach to Rendezvous Numbers vol.148, pp.4, 2006, https://doi.org/10.1007/s00605-006-0397-5
  5. Homogeneous polynomials and extensions of Hardy-Hilbert's inequality vol.285, pp.1, 2012, https://doi.org/10.1002/mana.201010035
  6. Non-linear plank problems and polynomial inequalities vol.30, pp.3, 2017, https://doi.org/10.1007/s13163-017-0220-y
  7. Rendezvous numbers in normed spaces vol.72, pp.03, 2005, https://doi.org/10.1017/S0004972700035255
  8. A geometric estimate on the norm of product of functionals vol.405, 2005, https://doi.org/10.1016/j.laa.2005.03.028
  9. Transfinite Diameter, Chebyshev Constant and Energy on Locally Compact Spaces vol.28, pp.3, 2008, https://doi.org/10.1007/s11118-008-9075-7
  10. POLARIZATION AND UNCONDITIONAL CONSTANTS OF 𝓟(2d*(1,ω)2) vol.29, pp.3, 2014, https://doi.org/10.4134/CKMS.2014.29.3.421
  11. Linear polarization constants of Hilbert spaces vol.300, pp.1, 2004, https://doi.org/10.1016/j.jmaa.2004.06.031
  12. Weak-closure and polarization constant by Gaussian measure vol.264, pp.2, 2010, https://doi.org/10.1007/s00209-009-0474-2