The real plank problem and some applications

K. Ball has proved the ``complex plank problem'': if $\left(x_{k}\right)_{k=1}^{n}$ is a sequence of norm $1$ vectors in a complex Hilbert space $\left(H, \langle \cdot,\cdot \rangle\right)$, then there exists a unit vector $x$ for which

$$\left\vert\langle{x}, x_{k}\rangle\right\vert\geq 1/\sqrt{n} ,\quad k=1, \ldots, n .$$

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

$$\left\vert\langle{x}, x_{k}\rangle\right\vert \geq \max\left\{\sqrt{\lambda_{1}/n}, 1/\sqrt{\lambda_{n}n}\right\} ,$$

where $\lambda_{1}$ is the smallest and $\lambda_{n}$ is the largest eigenvalue of the Hermitian matrix $A=\left[\langle{x_{j}}, x_{k}\rangle\right]$, $j, k=1, \ldots, n$. We have also improved known estimates for the norms of homogeneous polynomials which are products of linear forms on real Hilbert spaces.