g$ and the induced area element from $\bar g$ respectively. $\nu$ is the normal vector field of $\S$ and $\phi_\nu=\nb_\nu\phi$ is the normal derivative of $\phi$. $h(X,Y)=\<\nb_X \nu, Y\>$ is the classical second fundamental form of $\S$ and $H= \tr_{\bar g} h$ is the usual mean curvature of $\S$.

With the Reilly formula, some classical geometric inequalities can be readily proved.

[**Heintze-Karcher inequality:**]{} If $Ric\ge 0$ and $\S$ is mean convex, i.e., $H>0$, then $$\begin{aligned}
\label{HK0}
\int_\S \frac{1}{H}dA\ge  \frac{n}{n-1} Vol(\O).\end{aligned}$$

[**Minkowski inequality:**]{} If $Ric\ge 0$ and $\S$ is convex, i.e., $h \ge 0$, then $$\begin{aligned}
\label{M0}
Area(\S)^2 \ge  \frac{n}{n-1} Vol(\O)\int_\S H dA.\end{aligned}$$

[**Lichnerowicz inequality:**]{} If $Ric\ge (n-1)\bar g$ and $\S$ is empty, then $$\begin{aligned}
\label{L0}
\l_1(\bd)\ge n.\end{aligned}$$

Similar inequalities like were first derived by Heintze-Karcher [@HK] using the classical approach of Jacobian fields from Riemanian geometry. Ros [@Ros] proved the current form of this inequality using Reilly’s formula. Inequality was first derived by Minkowski [@Min] in the Euclidean case as a consequence of the famous Brunn-Minkowski theorem in convex geometry. Reilly [@Re3] proved this inequality under the condition of nonnegative Ricci by using his formula. Recently, Wang-Zhang [@WZ] gave an alternative proof of Minkowski inequality (\[M0\]) using ABP method. Inequality 