Some geometric inequalities by the ABP method

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 $\mathrm{\Sigma }$ of ${\mathbb{S}}^{n+m}$. As a consequence, it recovers the classical result that $\mathrm{\mid }{\mathbb{S}}^n\mathrm{\mid }\le \mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }$ 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.