  Let $\Omega_n$ stand for the volume of the unit ball in $\mathbb{R}^n$ for
$n\in\mathbb{N}$. In the present paper, we prove that the sequence
$\Omega_{n}^{1/(n\ln n)}$ is logarithmically convex and that the sequence
$\frac{\Omega_{n}^{1/(n\ln n)}}{\Omega_{n+1}^{1/[(n+1)\ln(n+1)]}}$ is strictly
decreasing for $n\ge2$. In addition, some monotonic and concave properties of
several functions relating to $\Omega_{n}$ are extended and generalized.
