LetD be the class of domains\(D_{a,p} = \left\{ {z \in \mathbb{C}^n |\left| {z_1 } \right|^{2_{p1} } + 2a\left| {z_1 } \right|^{P1} \left| {z_2 } \right|^{_{P2} } + \left| {z_2 } \right|^{2_{P2} } + \sum\limits_{j = 3}^n {\left| {z_j } \right|^{2_{Pj} }< 1} } \right\}\) forn≥2,a≥0 and p=(p1,...,n) ∈ (ℝ+) n such thatD a,p is convex. The classD is a class of convex bounded Reinhardt domains of ℂ n which are a generalization of complex ellipsoids. In this paper we compare Caratheodory balls and norm balls of the domainsD∈D. We prove that in this case a Caratheodory ball inD∈D is a norm ball if, and only if,D is a complex ellipsoid\(\left\{ {z \in \mathbb{C}\left| {\Sigma _{j = 1}^n \left| {z_j } \right|^{2_{Pj} }< 1} \right.} \right\}\) such thatp k=1 for exactly onek∈{1,…,n},p j=1/2 for allj≠k and the centre lies on thez k-axis.