| Summary: | 本文中我們將證明Kobayashi擬度量在凸域中的三角不等式成立，任一C<sup>n</sup>中不包含複仿射線之凸域皆可解析嵌入n維單位多重圓板，在凸域中的Carathéodory距離函數產生原來的拓樸以及在凸域中的hyperbolicity和measure hyperbolicity是等價的概念，進而推論到任一體積有限的凸域必須是hyperbolic，因此，當然是measure hyperbolic。
=== In this thesis , we prove that the triangle inequality of the Kobayashi pseudometric holds in any convex domain. Also , for a convex domain Q containing no complex affine line , we prove that Ω is biholomorphic to a subdomain of the unit polydisc D<sup>n</sup> and the topology induced by the Carathéodory distance function coincides with the Euclidean topology of Ω. Finally , we prove that hyperbolicity and measure hyperbolicity in a convex domain are equivalent. Moreover, any convex domain with finite Euclidean volume must be hyperbolic, therefore , it is measure hyperbolic.
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