  The problem of low rank approximation is ubiquitous in science. Traditionally
this problem is solved in unitary invariant norms such as Frobenius or spectral
norm due to existence of efficient methods for building approximations.
However, recent results reveal the potential of low rank approximations in
Chebyshev norm, which naturally arises in many applications. In this paper we
tackle the problem of building optimal rank-1 approximations in the Chebyshev
norm. We investigate the properties of alternating minimization algorithm for
building the low rank approximations and demonstrate how to use it to construct
optimal rank-1 approximation. As a result we propose an algorithm that is
capable of building optimal rank-1 approximations in Chebyshev norm for small
matrices.
