The authors are interested in computing the Carathéodory pseudodistance \(c_ G(0,z)\) from the center 0 of a balanced (i.e., complete circular) domain of holomorphy G in \({\mathbb{C}}^ n\) to a point \(z\in G\). When G is convex these distances are given by the well-known formula \[ c_ G(0,z)=\tanh^{-1} \inf \{\lambda >0:\quad z\in \lambda G\}, \] but this formula fails when G is nonconvex. The authors derive formula for these distances on certain (possibly nonconvex) Reinhardt domains of special types. They illustrate the difficulties in the general case by expliciting computing these distances for some points of the domain \[ G=\{z\in {\mathbb{C}}^ 2:\quad | z_ 1| <1,\quad | z_ 2| <1,\quad | z_ 1z_ 2| <1/2\}. \] The authors also give conditions on G that imply the product formula \[ c_{G\times D}((0,w'),(z,w''))=\max \{c_ G(0,z),c_ D(w',w'')\} \] for all domains D and all points w',w''\(\in D\). It apparently remains an open question as to whether the product formula holds in general.