K. Ball has proved the “complex plank problem”: if
(
x
k
)
k
=
1
n
\left (x_{k}\right )_{k=1}^{n}
is a sequence of norm
1
1
vectors in a complex Hilbert space
(
H
,
⟨
⋅
,
⋅
⟩
)
\left (H, \, \langle \cdot ,\cdot \rangle \right )
, then there exists a unit vector
x
x
for which
\[
|
⟨
x
,
x
k
⟩
|
≥
1
/
n
,
k
=
1
,
…
,
n
.
\left |\langle {x}, x_{k}\rangle \right |\geq 1/\sqrt {n}\,,\quad k=1, \ldots , n\,.
\]
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
x
we have derived the estimate
\[
|
⟨
x
,
x
k
⟩
|
≥
max
{
λ
1
/
n
,
1
/
λ
n
n
}
,
\left |\langle {x}, x_{k}\rangle \right | \geq \max \left \{\sqrt {\lambda _{1}/n},\, 1/\sqrt {\lambda _{n}n}\right \}\,,
\]
where
λ
1
\lambda _{1}
is the smallest and
λ
n
\lambda _{n}
is the largest eigenvalue of the Hermitian matrix
A
=
[
⟨
x
j
,
x
k
⟩
]
A=\left [\langle {x_{j}}, x_{k}\rangle \right ]
,
j
,
k
=
1
,
…
,
n
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.