Abstract
We study lower bounds for the norm of the product of polynomials and their applications to the so called plank problem. We are particularly interested in polynomials on finite dimensional Banach spaces, in which case our results improve previous works for large numbers of polynomials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arias-de-Reyna, J.: Gaussian variables, polynomials and permanents. Linear Algebra Appl. 285, 107–114 (1998)
Ball, K.M.: The plank problem for symmetric bodies. Invent. Math. 104, 535–543 (1991)
Ball, K.M.: The complex plank problem. Bull. London Math. Soc. 33, 433–442 (2001)
Bang, T.: A solution of the plank problem. Proc. Am. Math. Soc. 2, 990–993 (1951)
Beauzamy, B., Bombieri, E., Enflo, P., Montgomery, H.L.: Products of polynomials in many variables. J. Number Theory 36(2), 219–245 (1990)
Benítez, C., Sarantopoulos, Y., Tonge, A.: Lower bounds for norms of products of polynomials. Math. Proc. Camb. Philos. Soc. 124(3), 395–408 (1998)
Boyd, C., Ryan, R.: The norm of the product of polynomials in infinite dimensions. Proc. Edinb. Math. Soc. (Ser. 2) 49(1), 17–28 (2006)
Brudnyi, Yu., Ganzburg, M.: On an extremal problem for polynomials of n variables. Math. USSR-Izv. 37, 344–355 (1973)
Carando, D., Pinasco, D., Rodríguez, J.T.: Lower bounds for norms of products of polynomials on \(L_p\) spaces. Studia Math. 214, 157–166 (2013)
Kavadjiklis, A., Kim, S.G.: Plank type problems for polynomials on Banach spaces. J. Math. Anal. Appl. 396(2), 528–535 (2012)
Malicet, D., Nourdin, I., Peccati, G., Poly, G.: Squared chaotic random variables: new moment inequalities with applications. J. Funct. Anal. 270(2), 649–670 (2016)
Pappas, A., Révész, S.G.: Linear polarization constants of Hilbert spaces. J. Math. Anal. Appl. 300(1), 129–146 (2004)
Pinasco, D.: Lower bounds for norms of products of polynomials via Bombieri inequality. Trans. Am. Math. Soc. 364, 3993–4010 (2012)
Révész, S.G., Sarantopoulos, Y.: Plank problems, polarization and Chebyshev constants. Satellite conference on infinite dimensional function theory. J. Korean Math. Soc. 41(1), 157–174 (2004)
Rodríguez, J.T.: On the norm of products of polynomials on ultraproduct of Banach spaces. J. Math. Anal. Appl. 421(2), 805–816 (2015)
Rodríguez, J.T.: Desigualdades polinomiales en espacios de Banach. Ph.D .Thesis, Universidad de Buenos Aires (2016)
Tarski, A.: O stopniu równowa\(\dot{{\rm o}}\)ności wieloka̧tów. (English: On the degree of equivalence of polygons). Młody Matematyk 1, 37–44 (1931)
Tarski, A.: Uwagi o stopniu równowa\(\dot{{\rm o}}\)ności wieloka̧tów. (English: Remarks on the degree of equivalence of polygons). Parametr 2, 310–314 (1932)
Acknowledgements
This project was supported in part by UBACyT 20020130100474BA, PIP 11220130100329CO (CONICET) (2014–2016) and PICT 2015–2299.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Carando, D., Pinasco, D. & Rodríguez, J.T. Non-linear plank problems and polynomial inequalities. Rev Mat Complut 30, 507–523 (2017). https://doi.org/10.1007/s13163-017-0220-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13163-017-0220-y