  We propose a unique way how to choose a new inner product in a Hilbert space
with respect to which an originally non-self-adjoint operator similar to a
self-adjoint operator becomes self-adjoint. Our construction is based on
minimising a 'Hilbert-Schmidt distance' to the original inner product among the
entire class of admissible inner products. We prove that either the minimiser
exists and is unique, or it does not exist at all. In the former case we derive
a system of Euler-Lagrange equations by which the optimal inner product is
determined. A sufficient condition for the existence of the unique minimally
anisotropic metric is obtained. The abstract results are supplied by examples
in which the optimal inner product does not coincide with the most popular
choice fixed through a charge-like symmetry.
