PLANK PROBLEMS, POLARIZATION AND CHEBYSHEV CONSTANTS

Title & Authors
PLANK PROBLEMS, POLARIZATION AND CHEBYSHEV CONSTANTS
Revesz, Szilard-Gy.; Sarantopoulos, Yannis;

Abstract
In this work we discuss "plank problems" for complex Banach spaces and in particular for the classical $\small{L^{p}(\mu)}$ spaces. In the case $\small{1\;{\leq}\;p\;{\leq}\;2}$ 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
Plank problem;homogeneous polynomials over normed spaces;linear polarization constants;quasi-monotonous sequences;Banach-Mazur distance;characterization of Banach spaces;local theory of Banach spaces;com-plexification of Banach spaces;weak-star convergent subsequences;
Language
English
Cited by
1.
Numerical Plank Problem,;

Kyungpook mathematical journal, 2010. vol.50. 2, pp.289-295
2.
POLARIZATION AND UNCONDITIONAL CONSTANTS OF 𝓟(2d*(1,ω)2),;

대한수학회논문집, 2014. vol.29. 3, pp.421-428
1.
Homogeneous polynomials and extensions of Hardy-Hilbert's inequality, Mathematische Nachrichten, 2012, 285, 1, 47
2.
Weak-closure and polarization constant by Gaussian measure, Mathematische Zeitschrift, 2010, 264, 2, 459
3.
Potential Theoretic Approach to Rendezvous Numbers, Monatshefte für Mathematik, 2006, 148, 4, 309
4.
Transfinite Diameter, Chebyshev Constant and Energy on Locally Compact Spaces, Potential Analysis, 2008, 28, 3, 241
5.
Numerical Plank Problem, Kyungpook mathematical journal, 2010, 50, 2, 289
6.
Linear polarization constants of Hilbert spaces, Journal of Mathematical Analysis and Applications, 2004, 300, 1, 129
7.
A geometric estimate on the norm of product of functionals, Linear Algebra and its Applications, 2005, 405, 304
8.
Rendezvous numbers in normed spaces, Bulletin of the Australian Mathematical Society, 2005, 72, 03, 423
9.
Plank type problems for polynomials on Banach spaces, Journal of Mathematical Analysis and Applications, 2012, 396, 2, 528
10.
POLARIZATION AND UNCONDITIONAL CONSTANTS OF 𝓟(2d*(1,ω)2), Communications of the Korean Mathematical Society, 2014, 29, 3, 421
11.
The dth linear polarization constant of Rd, Journal of Functional Analysis, 2008, 255, 10, 2861
References
1.
Polarization constants for products of linear functionals over $\mathbb{R}^2$ and $\mathbb{C}^2$ and Chebyshev constants of the unit sphere, 2002.

2.
Linear Algebra Appl., 1998. vol.285. pp.107-114

3.
Invent. Math., 1991. vol.104. pp.535-543

4.
Bull. London Math. Soc., 2001. vol.33. pp.433-442

5.
Proc. Amer. Math. Soc., 1951. vol.2. pp.990-993

6.
Math. Proc. Cambridge Philos. Soc., 1998. vol.124. pp.395-408

7.
Springer Monographs in Mathematics, 1999.

8.
Math. Z., 1923. vol.17. pp.228-249

9.
Summing and nuclear norms in Banach space theory, 1987.

10.
Courant Aniversary Volume, 1948. pp.187-204

11.
Private communication, 2002.

12.
Bull. London Math. Soc., 1999. vol.31. pp.269-278

13.
Studia Math., 1978. vol.63. pp.207-212

14.
J. Math. Mech., 1966. vol.16. pp.127-134

15.
Inequalities: selecta of Elliot H. Lieb., 2002.

16.
The university of Texas at Austin, Txas Functional Analysis Seminar, 1983. pp.167-176

17.
Mathematika, 1960. vol.7. pp.98-100

18.
Bull. Amer. Math. Soc., 1963. vol.69. pp.494-496

19.
Proc. Amer. Math. Soc., 1964. vol.15. pp.967-973

20.
Private communication, 1996. pp.967-973

21.
Trans. Amer. Math. Soc., 1962. vol.104. pp.510-515

22.
Amer. Math. Monthly, 1965. vol.72. pp.577-591

23.
Funktsional Anal. i Prilozhen Appl., 1971. vol.5. pp.28-37

24.
Engl. transl.: Funct Anal. Appl., 1971. vol.5. pp.288-295

25.
Studia Math., 1999. vol.134. pp.1-33

26.
The volume of convex bodies and Banach space geometry, 1989.

27.
Problems and Theorems in Analysis I(Reprint of the 1st ed), 1972.

28.
Math. Proc. Cambridge Philos. Soc., 1986. vol.99. pp.263-271

29.
Ph.D. thesis. Brunel University, 1987.

30.
J. Math. Anal. Appl., 1998. vol.221. pp.698-711

31.
Tohoku Math. J., 1938. vol.44. pp.302-318

32.
Private communication, 0000.

33.
J. Australian Math. Soc. Ser., 1989. vol.A47. pp.466-482

34.
Cambridge stud. Adv. Math., 1991. vol.25.