Journal of the
Korean Mathematical Society JKMS

In this work we discuss ``plank problems" for $complex$ Banach spaces and in particular for the classical $L_{p}( \mu )$ spaces. In the case $1 \leq p \leq 2$ 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, complexification of Banach spaces, weak-star con

MSC numbers: Primary 46G25; Secondary 52A40, 46B07