  We study an eigenvalue problem for prescribed $\sigma_k$-curvature equations
of star-shaped, $k$-convex, closed hypersurfaces. We establish the existence of
a unique eigenvalue and its associated hypersurface, which is also unique,
provided that the given data is even. Moreover, we show that the hypersurface
must be strictly convex. A crucial aspect of our proof involves deriving
uniform estimates in $p$ for $L_p$-type prescribed curvature equations.
