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We show that on a certain class of bounded, complete Reinhardt domains in$\mathbb{C}^n$ that enjoy a lot of symmetries, the Carath\'eodorypseudo-distance and the geodesic distance of the complete K\"ahler-Einsteinmetric with Ricci curvature $-1$ are different.

Comparing the Carath\'eodory Pseudo-Distance and the K\"ahler-Einstein Distance on Complete Reinhardt Domains