It is shown that for n ≥ 2 and p > 2, where p is not an even integer, the only balls in the Carathéodory distance on $H_{p,n}=z\in {ℂ}^n:\parallel z{\parallel }_p<1$ which are balls with respect to the complex $l_p$ norm in ${ℂ}^n$ are those centered at the origin.