  We provide a class of geometric convex domains on which the
Carath\'eodory-Reiffen metric, the Bergman metric, the complete
K\"ahler-Einstein metric of negative scalar curvature are uniformly equivalent,
but not proportional to each other. In a two-dimensional case, we provide a
full description of curvature tensors of the Bergman metric on the weakly
pseudoconvex boundary point and show that invariant metrics are proportional to
each other if and only if the geometric convex domain is the Poincar\'e-disk.
