  By adapting the test functions introduced by Choi-Daskaspoulos \cite{c-d} and
Brendle-Choi-Daskaspoulos \cite{b-c-d} and exploring properties of the $k$-th
elementary symmetric functions $\sigma_{k}$ intensively, we show that for any
fixed $k$ with $1\leq k\leq n-1$, any strictly convex closed hypersurface in
$\mathbb{R}^{n+1}$ satisfying $\sigma_{k}^{\alpha}=\langle X,\nu \rangle$, with
$\alpha\geq \frac{1}{k}$, must be a round sphere. In fact, we prove a
uniqueness result for any strictly convex closed hypersurface in
$\mathbb{R}^{n+1}$ satisfying $F+C=\langle X,\nu \rangle$, where $F$ is a
positive homogeneous smooth symmetric function of the principal curvatures and
$C$ is a constant.
