  In this paper, we apply the so-called Alexandrov-Bakelman-Pucci (ABP) method
to establish some geometric inequalities. We first prove a logarithmic Sobolev
inequality for closed $n$-dimensional minimal submanifolds $\Sigma$ of $\mathbb
S^{n+m}$. As a consequence, it recovers the classical result that $|\mathbb
S^n| \leq |\Sigma|$ for $m = 1,2$. Next, we prove a Sobolev-type inequality for
positive symmetric two-tensors on smooth domains in $\mathbb R^n$ which was
established by D. Serre when the domain is convex. In the last application of
the ABP method, we formulate and prove an inequality related to
quermassintegrals of closed hypersurfaces of the Euclidean space.
