  We discuss some geometric problems related to the definitions of quasilocal
mass proposed by Brown-York \cite{BYmass1} \cite{BYmass2} and Liu-Yau
\cite{LY1} \cite{LY2}. Our discussion consists of three parts. In the first
part, we propose a new variational problem on compact manifolds with boundary,
which is motivated by the study of Brown-York mass. We prove that critical
points of this variation problem are exactly static metrics. In the second
part, we derive a derivative formula for the Brown-York mass of a smooth family
of closed 2 dimensional surfaces evolving in an ambient three dimensional
manifold. As an interesting by-product, we are able to write the ADM mass
\cite{ADM61} of an asymptotically flat 3-manifold as the sum of the Brown-York
mass of a coordinate sphere $S_r$ and an integral of the scalar curvature plus
a geometrically constructed function $\Phi(x)$ in the asymptotic region outside
$S_r $. In the third part, we prove that for any closed, spacelike, 2-surface
$\Sigma$ in the Minkowski space $\R^{3,1}$ for which the Liu-Yau mass is
defined, if $\Sigma$ bounds a compact spacelike hypersurface in $\R^{3,1}$,
then the Liu-Yau mass of $\Sigma$ is strictly positive unless $\Sigma$ lies on
a hyperplane. We also show that the examples given by \'{O} Murchadha, Szabados
and Tod \cite{MST} are special cases of this result.
