The Symmetrized Polydisc Cannot Be Exhausted by Domains Biholomorphic to Convex Domains

We prove that the symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains. Let D be the unit disc in C. Let σn = (σn,1, . . . , σn,n) : C n → C be defined as follows: σn,k(z1, . . . , zn) = ∑ 1≤j1<···<jk≤n zj1 . . . zjk , 1 ≤ k ≤ n. The set Gn = σn(D ) is called the symmetrized n-disc. The symmetrized bidisc G2 is the first… CONTINUE READING