In the problem of (simultaneous) Diophantine approximation in ${ℝ}^3$ (in the spirit of Hurwitz’s theorem), lower bounds for the critical determinant of the special three-dimensional body $$K_2:(y^2+z^2)(x^2+y^2+z^2)\le 1$$ play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant $\Delta \left(K_c\right)$ of more general star bodies $$K_c:{(y^2+z^2)}^{c/2}(x^2+y^2+z^2)\le 1,$$ where $c$ is any positive constant. These are obtained by inscribing into $K_c$ either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of $c$.