1 Lempert functions and Kobayashi metrics ...................... 6
1.1 Synopsis ................................................ 6
1.2 Lempert functions and their "derivatives" ............... 7
1.3 Balanced domains ....................................... 13
1.4 Kobayashi-Buseman metric ............................... 17
1.5 Interpolation in the Arakelian theorem ................. 19
1.6 Generalized Lempert function ........................... 21
1.7 Product property ....................................... 24
2 The symmetrized polydisc and the spectral ball .............. 27
2.1 Synopsis ............................................... 27
2.2 Preliminaries .......................................... 31
2.3 Cyclic matrices ........................................ 34
2.4.  n is not a Lu Qi-Keng domain for n ≥ 3 ................ 35
2.5 Generalized balanced domains ........................... 38
2.6 Notions of complex convexity ........................... 42
2.7 n   for n ≥ 3 ....................................... 47
2.8 Estimates for γ2n+1 (0; ℮2) ........................... 52
2.9 Continuity of ℓΩn (A,•) ................................ 58
2.10 Zeroes of kΩn .......................................... 61
2.11 The Kobayashi metric vs. the Lempert function .......... 66
3 Estimates and boundary behavior of invariant metrics on
C-convex domains ............................................ 69
3.1 Synopsis ............................................... 69
3.2 Estimates for the Caratheodory and Kobayashi metrics ... 71
3.3 Types of boundary points ............................... 73
3.4 Estimates for the Bergman kernel and the Bergman
metric ................................................. 76
3.5 Maximal basis. A counterexample ........................ 81
3.6 Estimates in a maximal basis ........................... 83
3.7 Localizations .......................................... 85
3.8 Localization of the Bergman kernel and the Bergman
metric ................................................. 88
3.9 Boundary behavior of invariant metrics of planar
domains ................................................ 93
References ..................................................... 95
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