A note on Costara's paper

Abstract

Abstract. We show that the symmetrized bidisc2 cannot be exhausted by domains biholomorphic to convex domains.

Discontinuity of the Lempert Function and the Kobayashi-Royden Metric of the Spectral Ball

“…By (3), the equality κ Ωn (A; B) is equivalent to E * ,A (B) = 0. By property (8) in Proposition 3 and its proof, we have a matrix Y ∈ M n such that −Y A + AY = B. Then the mapping λ → e −λY Ae λY satisfies all the required properties.…”

Schwarz lemma for the tetrablock

“…[3], [4], [7], [16], [11], [5] and others). It was shown that G 2 cannot be exhausted by domains biholomorphic to convex ones (see [7], [10]). So symmetrized bidisc delivers a counterexample to the above problem.…”

Balanced domains and convexity

“…Let us show the implication (3) =⇒ (4). Suppose that Φ ν : D ν → G ν is a biholomorphic mapping and that G ν is convex.…”

“…It is known that on G 2 holomorphically invariant distances are equal (see [1][2][3]), G 2 is not convex (see [2]), G 2 cannot be exhausted by domains which are biholomorphic to convex domains (see [4]), G 2 can be exhausted by domains which are strictly linearly convex (see [10]). Note that G 2 is a (1,2)-balanced pseudoconvex domain.…”