Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 54: Issue 3
Extreme 2-homogeneous polynomials on the plane with a hexagonal norm and applications to the polarization and unconditional constants
Author:
Sung Guen Kim Kyungpook National University, Daegu 702-701, South Korea

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Pages:
362–393
Online Publication Date:
Sep 2017
Publication Date:
01 Sep 2017
Article Category:
Research Article
DOI:
https://doi.org/10.1556/012.2017.54.3.1371
Keywords:
The Krein-Milman Theorem; extreme points; 2-homogeneous polynomials; the plane with a hexagonal norm; the polarization and unconditional constants
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We classify the extreme 2-homogeneous polynomials on ℝ2 with the hexagonal norm of weight ½. As applications, using its extreme points with the Krein-Milman Theorem, we explicitly compute the polarization and unconditional constants of P(2ℝh(12)2).

  • [1]

    Aron, R.M. and Klimek, M., Supremum norms for quadratic polynomials, Arch. Math. (Basel), 76 (2001), 73–80.

  • [2]

    Choi, Y.S., Ki, H. and Kim, S.G., Extreme polynomials and multilinear forms on l1, J. Math. Anal. Appl. 228 (1998), 467–482.

  • [3]

    Choi, Y.S. and Kim, S.G., The unit ball of P ( 2 l 2 2 ), Arch. Math. (Basel) 71 (1998), 472–480.

  • [4]

    Choi, Y.S. and Kim, S.G., Extreme polynomials on c0, Indian J. Pure Appl. Math., 29 (1998), 983–989.

  • [5]

    Choi, Y.S. and Kim, S.G., Smooth points of the unit ball of the space ρ(2 l1), Results Math., 36 (1999), 26–33.

  • [6]

    Choi, Y.S. and Kim, S.G., Exposed points of the unit balls of the spaces P ( 2 l p 2 ) (p = 1, 2, ∞), Indian J. Pure Appl. Math., 35 (2004), 37–41.

    • Search Google Scholar
    • Export Citation
  • [7]

    Dineen, S. , Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag, London (1999).

  • [8]

    Gamez-Merino, J.L., Munoz-Fernandez, G.A., Sanchez, V.M. and Seoane-Sepulveda, J.B., Inequalities for polynomials on the unit square via the Krein-Milman Theorem, J. Convex Anal., 340 (2013), no. 1, 125–142.

    • Search Google Scholar
    • Export Citation
  • [9]

    Grecu, B.C. , Geometry of three-homogeneous polynomials on real Hilbert spaces, J. Math. Anal. Appl., 246 (2000), 217–229.

  • [10]

    Grecu, B.C. , Smooth 2-homogeneous polynomials on Hilbert spaces, Arch. Math., (Basel) 76 (2001), no. 6, 445–454.

  • [11]

    Grecu, B.C. , Geometry of 2-homogeneous polynomials on lp spaces, 1 < p < ∞, J. Math. Anal. Appl., 273 (2002), 262–282.

  • [12]

    Grecu, B.C. , Extreme 2-homogeneous polynomials on Hilbert spaces, Quaest. Math., 25 (2002), no. 4, 421–435.

  • [13]

    Grecu, B.C. , Geometry of homogeneous polynomials on two-dimensional real Hilbert spaces J. Math. Anal. Appl., 293 (2004), 578–588.

  • [14]

    Grecu, B.C., G.A. Munoz-Fernandez, and J.B. Seoane-Sepulveda, The unit ball of the complex P(3 H), Math. Z., 263 (2009), 775–785.

  • [15]

    Kim, S.G. , Exposed 2-homogeneous polynomials on P ( 2 l p 2 ) (1 ≤ p ≤ ∞), Math. Proc. Royal Irish Acad., 107 (2007), 123–129.

  • [16]

    Kim, S.G. , The unit ball of ℒ s ( 2 l ∞ 2 ), Extracta Math., 24 (2009), 17–29.

  • [17]

    Kim, S.G. , The unit ball of ρ(2 d*(1, w)2), Math. Proc. Royal Irish Acad., 111 (2) (2011), 79–94.

  • [18]

    Kim, S.G. , The unit ball of ℒs(2 d*(1, w)2), Kyungpook Math. J., 53 (2013), 295–306.

  • [19]

    Kim, S.G. , Smooth polynomials of ρ(2 d*(1, w)2), Math. Proc. Royal Irish Acad., 113A (1) (2013), 45–58.

  • [20]

    Kim, S.G. , Extreme bilinear forms of ℒ(2 d*(1, w)2), Kyungpook Math. J., 53 (2013), 625–638.

  • [21]

    Kim, S.G. , Exposed symmetric bilinear forms of ℒs(2 d*(1, w)2), Kyungpook Math. J., 54 (2014), 341–347.

  • [22]

    Kim, S.G. , Polarization and unconditional constants of ρ(2 d*(1, w)2), Commun. Korean Math. Soc., 29 (2014), 421–428.

  • [23]

    Kim, S.G. , Exposed 2-homogeneous polynomials on the two-dimensional real predual of Lorentz sequence space, Mediterr. J. Math., 13 (2016), 2827–2839.

    • Search Google Scholar
    • Export Citation
  • [24]

    Kim, S.G. , Extremal problems for ℒ s ( 2 ℝ h ( w ) 2 ), Kyungpook Math. J., 57 (2) (2017), 223–232.

  • [25]

    Kim, S.G. and Lee, S.H., Exposed 2-homogeneous polynomials on Hilbert spaces, Proc. Amer. Math. Soc., 131 (2003), 449–453.

  • [26]

    Kim, S.G. , Exposed 2-homogeneous polynomials on the plane with a hexagonal norm, Preprint.

  • [27]

    Kim, S.G. , Smooth 2-homogeneous polynomials on the plane with a hexagonal norm, Preprint.

  • [28]

    Kim, S.G. , The unit ball of ℒ s ( 2 l ∞ 3 ), Comment. Math. Prace Mat., 57 (2017), 1–7.

  • [29]

    Kim, S.G. , The unit ball of ℒ ( 2 ℝ h ( w ) 2 ), Bull. Korean Math. Soc., 54 (2) (2017), 417–428.

  • [30]

    Konheim, A.G. and Rivlin, T.J., Extreme points of the unit ball in a space of real polynomials, Amer. Math. Monthly 73 (1966), 505–507.

    • Search Google Scholar
    • Export Citation
  • [31]

    Milev, L. and Naidenov, N., Strictly definite extreme points of the unit ball in a polynomial space, C. R. Acad. Bulg. Sci., 61 (2008), 1393–1400.

    • Search Google Scholar
    • Export Citation
  • [32]

    Milev, L. and Naidenov, N., Semidefinite extreme points of the unit ball in a polynomial space, J. Math. Anal. Appl., 405 (2013), 631–641.

    • Search Google Scholar
    • Export Citation
  • [33]

    Munoz-Fernandez, G.A., Pellegrino, D., Seoane-Sepulveda, J.B. and Weber, A., Supremum norms for 2-homogeneous polynomials on circle sectors, J. Convex Anal., 21 (2014), no. 3, 745–764.

    • Search Google Scholar
    • Export Citation
  • [34]

    Munoz-Fernandez, G.A., Revesz, S. and Seoane-Sepulveda, J.B, Geometry of homogeneous polynomials on non symmetric convex bodies, Math. Scand., 105 (2009), 147–160.

    • Search Google Scholar
    • Export Citation
  • [35]

    Munoz-Fernandez, G.A. and Seoane-Sepulveda, J.B, Geometry of Banach spaces of trinomials, J. Math. Anal. Appl., 340 (2008), 1069–1087.

    • Search Google Scholar
    • Export Citation
  • [36]

    Neuwirth, S. , The maximum modulus of a trigonometric trinomial, J. Anal. Math., 104 (2008), 371–396.

  • [37]

    Revesz, S. and Sarantopoulos, Y., Plank problems, polarization and Chebyshev constants, J. Korean Math. Soc., 41 (2004), 157–174.

  • [38]

    Ryan, R.A. and Turett, B., Geometry of spaces of polynomials, J. Math. Anal. Appl., 221 (1998), 698–711.

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Cover Studia Scientiarum Mathematicarum Hungarica
Studia Scientiarum Mathematicarum Hungarica
Print ISSN:
0081-6906
Online ISSN:
1588-2896

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András STIPSICZ (Rényi Institute of Mathematics)
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Studia Scientiarum Mathematicarum Hungarica
Language English
French
German
Size B5
Year of
Foundation		 1966
Volumes
per Year		 1
Issues
per Year		 4
Founder Magyar Tudományos Akadémia  
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Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
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ISSN 0081-6906 (Print)
ISSN 1588-2896 (Online)

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