In this paper, we extend the Brown's fundamental theorem on fine ferromagnetic particles to the case of a general ellipsoid. By means of Poincar{\'e} inequality for the Sobolev space $H^1 (\Omega, \text{\tmtextsf{R}}^3)$, and some properties of the induced magnetic field operator, it is rigorously proven that for an ellipsoidal particle, with diameter $d$, there exists a critical size (diameter) $d_c$ such that for $d < d_c$ the uniform magnetization states are the only global minimizers of the Gibbs-Landau free energy functional $\mathcal{G}_{\mathcal{L}}$. A lower bound for $d_c$ is then given in terms of the demagnetizing factors.