Abstract
The study and classification of the extreme points of the unit ball of a Banach space is a classical problem in functional analysis. This question is particularly interesting in the case of Banach spaces of polynomials.
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G. Araújo, G.A. Muñoz-Fernández, D.L. RodríguezVidanes, J.B. Seoane-Sepúlveda, Sharp Bernstein inequalities using convex analysis techniques. Math. Inequal. Appl. 23(2), 725–750 (2020). https://doi.org/10.7153/mia-2020-23-61
R.M. Aron, M. Klimek, Supremum norms for quadratic polynomials. Arch. Math. (Basel) 76(1), 73–80 (2001). https://doi.org/10.1007/s000130050544
L. Bernal-González, G.A. Muñoz-Fernández, D.L. RodríguezVidanes, J.B. Seoane-Sepúlveda, A complete study of the geometry of 2-homogeneous polynomials on circle sectors, Preprint (2019)
C. Boyd, R.A. Ryan, Geometric theory of spaces of integral polynomials and symmetric tensor products. J. Funct. Anal. 179(1), 18–42 (2001)
C. Boyd, R.A. Ryan, N. Snigireva, Geometry of spaces of orthogonally additive polynomials on C(K). J. Geom. Anal. (2020). https://doi.org/10.1007/s12220-019-00240-0
Y.S. Choi, S.G. Kim, Extreme polynomials on c0. Indian J. Pure Appl. Math. 29(10), 983–989 (1998)
Y.S. Choi, S.G. Kim, H. Ki, Extreme polynomials and multilinear forms on l1. J. Math. Anal. Appl. 228(2), 467–482 (1998). https://doi.org/10.1006/jmaa.1998.6161
V. Dimant, D. Galicer, R. García, Geometry of integral polynomials, M-ideals and unique norm preserving extensions. J. Funct. Anal. 262(5), 1987–2012 (2012)
S. Dineen, Extreme integral polynomials on a complex Banach space. Math. Scand. 92(1), 129–140 (2003)
J.L. Gámez-Merino, G.A. Muñoz-Fernández, V.M. Sánchez, J.B. Seoane-Sepúlveda, Inequalities for polynomials on the unit square via the Krein-Milman theorem. J. Convex Anal. 20(1), 125–142 (2013)
B.C. Grecu, Extreme 2-homogeneous polynomials on Hilbert spaces. Quaest. Math. 25(4), 421–435 (2002). https://doi.org/10.2989/16073600209486027
B.C. Grecu, Geometry of three-homogeneous polynomials on real Hilbert spaces. J. Math. Anal. Appl. 246(1), 217–229 (2000). https://doi.org/10.1006/jmaa.2000.6783
B.C. Grecu, G.A. Muñoz-Fernández, J.B. Seoane-Sepúlveda, The unit ball of the complex \(\mathcal {P}({ }^{3}H)\). Math. Z. 263(4), 775–785 (2009). https://doi.org/10.1007/s00209-008-0438-y
P. Jiménez-Rodríguez, G.A. Muñoz-Fernández, D.L. RodríguezVidanes, Geometry of spaces of homogeneous trinomials on \(\mathbb {R}^{2}\). Banach J. Math. Anal. 15(4), Paper No. 61, 22 (2021). https://doi.org/10.1007/s43037-021-00144-8
S.G. Kim, The unit ball of \(\mathcal {P}({ }^{2}D_{\ast }(1,W)^{2})\). Math. Proc. R. Irish Acad. 111A(2), 79–94 (2011)
A.G. Konheim, T.J. Rivlin, Extreme points of the unit ball in a space of real polynomials. Am. Math. Mon. 73, 505–507 (1966). https://doi.org/10.2307/2315472
L. Milev, N. Naidenov, Semidefinite extreme points of the unit ball in a polynomial space. J. Math. Anal. Appl. 405(2), 631–641 (2013)
G.A. Muñoz-Fernández, D. Pellegrino, J.B. Seoane-Sepúlveda, A. Weber, Supremum norms for 2-homogeneous polynomials on circle sectors. J. Convex Anal. 21(3), 745–764 (2014)
G.A. Muñoz-Fernández, S. Gy Révész, J.B. Seoane-Sepúlveda, Geometry of homogeneous polynomials on non symmetric convex bodies. Math. Scand. 105(1), 147–160 (2009). https://doi.org/10.7146/math.scand.a-15111
G.A. Muñoz-Fernández, J.B. Seoane-Sepúlveda, Geometry of Banach spaces of trinomials. J. Math. Anal. Appl. 340(2), 1069–1087 (2008). https://doi.org/10.1016/j.jmaa.2007.09.010
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Ferrer, J., García, D., Maestre, M., Muñoz, G.A., Rodríguez, D.L., Seoane, J.B. (2022). Introduction. In: Geometry of the Unit Sphere in Polynomial Spaces. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-031-23676-1_1
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