Smooth 2-homogeneous polynomials on the plane with a hexagonal norm
Main Article Content
Abstract
Motivated by the classifications of extreme and exposed 2-homogeneous polynomials on the plane with the hexagonal norm ||(x, y)|| = max{|y|, |x| + |y|/2} (see [15, 16]), we classify all smooth 2-homogeneous polynomials on R2 with the hexagonal norm.
Downloads
Download data is not yet available.
Metrics
Metrics Loading ...
Article Details
How to Cite
Kim, S. G. (2022). Smooth 2-homogeneous polynomials on the plane with a hexagonal norm. Extracta Mathematicae, 37(2), 243-259. https://doi.org/10.17398/2605-5686.37.2.243
Issue
Section
Banach Spaces
References
[1] R.M. Aron, M. Klimek, Supremum norms for quadratic polynomials, Arch. Math. (Basel) 76 (2001), 73 – 80.
[2] Y.S. Choi, H. Ki, S.G. Kim, Extreme polynomials and multilinear forms on l1 , J. Math. Anal. Appl. 228 (1998), 467 – 482.
[3] Y.S. Choi, S.G. Kim, The unit ball of P(2l2 2 ), Arch. Math. (Basel) 71 (1998), 472 – 480.
[4] Y.S. Choi, S.G. Kim, Extreme polynomials on c0, Indian J. Pure Appl. Math. 29 (1998), 983 – 989.
[5] Y.S. Choi, S.G. Kim, Smooth points of the unit ball of the space P(2 l1 ), Results Math. 36 (1999), 26 – 33.
[6] Y.S. Choi, S.G. Kim, Exposed points of the unit balls of the spaces P(2 lp 2 ) (p = 1, 2, ∞), Indian J. Pure Appl. Math. 35 (2004), 37 – 41.
[7] S. Dineen, “Complex Analysis on Infinite Dimensional Spaces”, Springer-Verlag, London, 1999.
[8] B.C. Grecu, G.A. Muñoz-Fernández, J.B. Seoane-Sepúlveda, The unit ball of the complex P(3H), Math. Z. 263 (2009), 775 – 785.
[9] R.B. Holmes, “Geometric Functional Analysis and its Applications”, Springer-Verlag, New York-Heidelberg, 1975.
[10] S.G. Kim, Exposed 2-homogeneous polynomials on P(2 lp 2) for 1 ≤ p ≤ ∞), Math. Proc. R. Ir. Acad. 107 (2007), 123 – 129.
[11] S.G. Kim, The unit ball of P(2 D∗ (1, W )2 ), Math. Proc. R. Ir. Acad. 111A (2) (2011), 79 – 94.
[12] S.G. Kim, Smooth polynomials of P(2d∗ (1, w)2 ), Math. Proc. R. Ir. Acad. 113A (1) (2013), 45 – 58.
[13] S.G. Kim, Polarization and unconditional constants of P(2d∗ (1, w)2), Commun. Korean Math. Soc. 29 (2014), 421 – 428.
[14] S.G. Kim, Exposed 2-homogeneous polynomials on the two-dimensional real predual of Lorentz sequence space, Mediterr. J. Math. 13 (2016), 2827 – 2839.
[15] S.G. Kim, Extreme 2-homogeneous polynomials on the plane with a hexagonal norm and applications to the polarization and unconditional constants,Studia Sci. Math. Hungar. 54 (2017), 362 – 393.
[16] S.G. Kim, Exposed polynomials of P(2 Rh( 2 1 )), Extracta Math. 33(2) (2018), 2 127 – 143.
[17] S.G. Kim, S.H. Lee, Exposed 2-homogeneous polynomials on Hilbert spaces, Proc. Amer. Math. Soc. 131 (2003), 449 – 453.
[18] G.A. Muñoz-Fernández, S.Gy. Révész, J.B. Seoane-Sepúlveda, Geometry of homogeneous polynomials on non symmetric convex bodies, Math. Scand. 105 (2009), 147 – 160.
[19] R.A. Ryan, B. Turett, Geometry of spaces of polynomials, J. Math. Anal. Appl. 221 (1998), 698 – 711.
[2] Y.S. Choi, H. Ki, S.G. Kim, Extreme polynomials and multilinear forms on l1 , J. Math. Anal. Appl. 228 (1998), 467 – 482.
[3] Y.S. Choi, S.G. Kim, The unit ball of P(2l2 2 ), Arch. Math. (Basel) 71 (1998), 472 – 480.
[4] Y.S. Choi, S.G. Kim, Extreme polynomials on c0, Indian J. Pure Appl. Math. 29 (1998), 983 – 989.
[5] Y.S. Choi, S.G. Kim, Smooth points of the unit ball of the space P(2 l1 ), Results Math. 36 (1999), 26 – 33.
[6] Y.S. Choi, S.G. Kim, Exposed points of the unit balls of the spaces P(2 lp 2 ) (p = 1, 2, ∞), Indian J. Pure Appl. Math. 35 (2004), 37 – 41.
[7] S. Dineen, “Complex Analysis on Infinite Dimensional Spaces”, Springer-Verlag, London, 1999.
[8] B.C. Grecu, G.A. Muñoz-Fernández, J.B. Seoane-Sepúlveda, The unit ball of the complex P(3H), Math. Z. 263 (2009), 775 – 785.
[9] R.B. Holmes, “Geometric Functional Analysis and its Applications”, Springer-Verlag, New York-Heidelberg, 1975.
[10] S.G. Kim, Exposed 2-homogeneous polynomials on P(2 lp 2) for 1 ≤ p ≤ ∞), Math. Proc. R. Ir. Acad. 107 (2007), 123 – 129.
[11] S.G. Kim, The unit ball of P(2 D∗ (1, W )2 ), Math. Proc. R. Ir. Acad. 111A (2) (2011), 79 – 94.
[12] S.G. Kim, Smooth polynomials of P(2d∗ (1, w)2 ), Math. Proc. R. Ir. Acad. 113A (1) (2013), 45 – 58.
[13] S.G. Kim, Polarization and unconditional constants of P(2d∗ (1, w)2), Commun. Korean Math. Soc. 29 (2014), 421 – 428.
[14] S.G. Kim, Exposed 2-homogeneous polynomials on the two-dimensional real predual of Lorentz sequence space, Mediterr. J. Math. 13 (2016), 2827 – 2839.
[15] S.G. Kim, Extreme 2-homogeneous polynomials on the plane with a hexagonal norm and applications to the polarization and unconditional constants,Studia Sci. Math. Hungar. 54 (2017), 362 – 393.
[16] S.G. Kim, Exposed polynomials of P(2 Rh( 2 1 )), Extracta Math. 33(2) (2018), 2 127 – 143.
[17] S.G. Kim, S.H. Lee, Exposed 2-homogeneous polynomials on Hilbert spaces, Proc. Amer. Math. Soc. 131 (2003), 449 – 453.
[18] G.A. Muñoz-Fernández, S.Gy. Révész, J.B. Seoane-Sepúlveda, Geometry of homogeneous polynomials on non symmetric convex bodies, Math. Scand. 105 (2009), 147 – 160.
[19] R.A. Ryan, B. Turett, Geometry of spaces of polynomials, J. Math. Anal. Appl. 221 (1998), 698 – 711.