PLANK PROBLEMS, POLARIZATION AND CHEBYSHEV CONSTANTS Revesz, Szilard-Gy.; Sarantopoulos, Yannis;
In this work we discuss "plank problems" for complex Banach spaces and in particular for the classical spaces. In the case we obtain optimal results and for finite dimensional complex Banach spaces, in a special case, we have improved an early result by K. Ball . 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.
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;