  A new spinorial strategy for the construction of geometric inequalities
involving the Arnowitt-Deser-Misner (ADM) mass of black hole systems in general
relativity is presented. This approach is based on a second order elliptic
equation (the approximate twistor equation) for a valence 1 Weyl spinor. This
has the advantage over other spinorial approaches to the construction of
geometric inequalities based on the Sen-Witten-Dirac equation that it allows to
specify boundary conditions for the two components of the spinor. This greater
control on the boundary data has the potential of giving rise to new geometric
inequalities involving the mass. In particular, it is shown that the mass is
bounded from below by an integral functional over a marginally outer trapped
surface (MOTS) which depends on a freely specifiable valence 1 spinor. From
this main inequality, by choosing the free data in an appropriate way, one
obtains a new nontrivial bounds of the mass in terms of the inner expansion of
the MOTS. The analysis makes use of a new formalism for the $1+1+2$
decomposition of spinorial equations.
