  In the paper we study the geometric properties of a large family of domains,
called the generalized tetrablocks, related to the $\mu$-synthesis, containing
both the family of the symmetrized polydiscs and the family of the
$\mu_{1,n}$-quotients $\mathbb E_n$, $n\geq2$, introduced recently by G.
Bharali. It is proved that the generalized tetrablock cannot be exhausted by
domains biholomorphic to convex ones. Moreover, it is shown that the
Carath\'eodory distance and the Lempert function are not equal on a large
subfamily of the generalized tetrablocks, containing i.a. $\mathbb E_n$,
$n\geq4$. We also derive a number of geometric properties of the generalized
tetrablocks as well as the $\mu_{1,n}$-quotients. As a by-product, we get that
the pentablock, another domain related to the $\mu$-synthesis problem
introduced recently by J. Agler, Z. A. Lykova, and N. J. Young, cannot be
exhausted by domains biholomorphic to convex ones.
