Geometric properties of domains related to $\mu$-synthesis

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 ${\mathbb{E}}_n$, $n\ge 2$, 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. ${\mathbb{E}}_n$, $n\ge 4$. 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.