  In this paper, we investigate closed strictly convex hypersurfaces in
$\mathbb{R}^{n+1}$ which shrink self-similarly under a large family of fully
nonlinear curvature flows by high powers of curvature. When the speed function
is given by powers of a homogeneous of degree $1$ and inverse concave function
of the principal curvatures with power greater than $1$, we prove that the only
such hypersurfaces are round spheres. We also prove that slices are the only
closed strictly convex self-similar solutions to such curvature flows in the
hemisphere $\mathbb{S}^{n+1}_{+}$ with power greater than or equal to $1$.
