  A generalization of the Riemannian Penrose inequality in $n$-dimensional
space ($3\le n<8$) is done. We introduce a parameter $\alpha$
($-\frac{1}{n-1}<\alpha < \infty$) indicating the strength of the gravitational
field, and define a refined attractive gravity probe surface (refined AGPS)
with $\alpha$. Then, we show the area inequality for a refined AGPS, $A \le
\omega_{n-1} \left[ (n+2(n-1)\alpha)Gm /(1+(n-1)\alpha)
\right]^{\frac{n-1}{n-2}}$, where $A$ is the area of the refined AGPS,
$\omega_{n-1}$ is the area of the standard unit $(n-1)$-sphere, $G$ is Newton's
gravitational constant and $m$ is the Arnowitt-Deser-Misner mass. The obtained
inequality is applicable not only to surfaces in strong gravity regions such as
a minimal surface (corresponding to the limit $\alpha \to \infty$), but also to
those in weak gravity existing near infinity (corresponding to the limit
$\alpha \to -\frac{1}{n-1}$).
