Smooth 2-homogeneous polynomials on the plane with a hexagonal norm

Authors

  • Sung Guen Kim Department of Mathematics, Kyungpook National University Daegu 702-701, South Korea

DOI:

https://doi.org/10.17398/2605-5686.37.2.243

Keywords:

The Krein-Milman theorem, smooth points, extreme points, 2-homogeneous polynomials on the plane with the hexagonal norm

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.

References

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

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

Y.S. Choi, S.G. Kim, The unit ball of P(2l2 2 ), Arch. Math. (Basel) 71 (1998), 472 – 480.

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

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

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.

S. Dineen, “Complex Analysis on Infinite Dimensional Spaces”, Springer-Verlag, London, 1999.

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.

R.B. Holmes, “Geometric Functional Analysis and its Applications”, Springer-Verlag, New York-Heidelberg, 1975.

S.G. Kim, Exposed 2-homogeneous polynomials on P(2 lp 2) for 1 ≤ p ≤ ∞), Math. Proc. R. Ir. Acad. 107 (2007), 123 – 129.

S.G. Kim, The unit ball of P(2 D∗ (1, W )2 ), Math. Proc. R. Ir. Acad. 111A (2) (2011), 79 – 94.

S.G. Kim, Smooth polynomials of P(2d∗ (1, w)2 ), Math. Proc. R. Ir. Acad. 113A (1) (2013), 45 – 58.

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

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

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.

S.G. Kim, Exposed polynomials of P(2 Rh( 2 1 )), Extracta Math. 33(2) (2018), 2 127 – 143.

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

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.

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

Downloads

Published

2022-12-01

Issue

Section

Banach Spaces

How to Cite

Smooth 2-homogeneous polynomials on the plane with a hexagonal norm. (2022). Extracta Mathematicae, 37(2), 243-259. https://doi.org/10.17398/2605-5686.37.2.243