%0 Journal Article
%A Muñoz-Fernández, Gustavo A.
%A Sarantopoulos, Y
%A Seoane-Sepúlveda, Juan B.
%T The real plank problem and some applications.
%D 2010
%@ 0002-9939
%U https://hdl.handle.net/20.500.14352/43652
%X K. Ball has proved the "complex plank problem": if (x(k))(k=1)(n) is a sequence of norm I vectors in a complex Hilbert space (H, (., .)), then there exists a unit vector x for which |< x,x(k)>| >= 1/root n, k = 1,...,n. In general, this result is not true on real Hilbert spaces. However, in special cases we prove that the same result holds true. In general, for some unit vector x we have derived the estimate |< x,x(k)>| >= max{root lambda(1)/n, 1/root lambda(n)n}, where lambda(1) is the smallest and lambda(n) is the largest eigenvalue of the Hermitian matrix A = [(x(j), x(k))], j, k = 1,...,n. We have also improved known estimates for the norms of homogeneous polynomials which are products of linear forms on real Hilbert spaces.
%~