The authors construct a complex norm \(N^*\) that extends the real euclidean norm and prove that it is the smallest complex norm in the following sense: If N is any complex norm in \({\mathbb{C}}^ n\) which coincides with the real euclidean norm \(| \cdot |\) in \({\mathbb{R}}^ n\) and N(z)\(\leq | z|\) for \(z\in {\mathbb{C}}^ n\), then \(N^*(z)\leq N(z)\) for \(z\in {\mathbb{C}}^ n.\) Let \(B^*_ n\) be the unit ball with respect to \(N^*\). Then \(B_ n^*\) is a convex complete circular domain with only a continuous boundary for \(n>1\). The authors obtain some complex geometric properties of \(B^*_ n:\) \(B^*_ n\) is neither biholomorphically equivalent to the unit ball \(B^ n\) nor the polydisc \(\Delta^ n\). In fact, \(B^*_ n\) is not even homogeneous. In particular, if \(n=2\), \(B^*_ 2\) is biholomorphically equivalent to \(D=\{z\in {\mathbb{C}}^ 2:| z_ 1| +| z_ 2| <1\},\) a rigid domain studied earlier by \textit{N. Kritikos} [Math. Ann. 99, 321-341 (1928), JFM 54.0373.02].