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

TL;DR

This paper investigates the geometric structure of generalized tetrablocks related to $mbda$-synthesis, revealing their non-convexity and differences between Carathe9odory and Lempert metrics, with implications for related domains.

Contribution

It establishes that generalized tetrablocks cannot be approximated by convex domains and analyzes metric properties, extending understanding of domains linked to $mbda$-synthesis.

Findings

Generalized tetrablocks are not biholomorphically convex.

Carathe9odory distance and Lempert function differ on many tetrablocks.

The pentablock cannot be exhausted by convex domains.

Abstract

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.