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The Symmetrized Bidisc and Lempert's Theorem
Author(s) -
Costara C.
Publication year - 2004
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609304003200
Subject(s) - mathematics , bounded function , convex domain , domain (mathematical analysis) , mathematics subject classification , regular polygon , pure mathematics , convex set , open set , combinatorics , mathematical analysis , geometry , convex optimization
Let G ⊆ C 2 be the open symmetrized bidisc, namely G = {(λ 1 + λ 2 , λ 1 λ 2 ) : |λ 1 | < 1, |λ 2 | < 1}. In this paper, a proof is given that G is not biholomorphic to any convex domain in C 2 . By combining this result with earlier work of Agler and Young, the author shows that G is a bounded domain on which the Carathéodory distance and the Kobayashi distance coincide, but which is not biholomorphic to a convex set. 2000 Mathematics Subject Classification 32F45 (primary), 15A18 (secondary).