Uniqueness of closed self-similar solutions to $\sigma_k^{\alpha}$-curvature flow

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.