  We describe a new non-parametric technique for reconstructing the mass
distribution in galaxy clusters with strong lensing, i.e., from multiple images
of background galaxies. The observed positions and redshifts of the images are
considered as rigid constraints and through the lens (ray-trace) equation they
provide us with linear constraint equations. These constraints confine the mass
distribution to some allowed region, which is then found by linear programming.
Within this allowed region we study in detail the mass distribution with
minimum mass-to-light variation; also some others, such as the smoothest mass
distribution. The method is applied to the extensively studied cluster Abell
370, which hosts a giant luminous arc and several other multiply imaged
background galaxies. Our mass maps are constrained by the observed positions
and redshifts (spectroscopic or model-inferred by previous authors) of the
giant arc and multiple image systems. The reconstructed maps obtained for \a370
reveal a detailed mass distribution, with substructure quite different from the
light distribution. The method predicts the bimodal nature of the cluster and
that the projected mass distribution is indeed elongated along the axis defined
by the two dominant cD galaxies. But the peaks in the mass distribution appear
to be offset from the centres of the cDs. We also present an estimate for the
total mass of the central region of the cluster. This is in good agreement with
previous mass determinations. The total mass of the central region is
M=(2.0-2.7) 10^14 Msun/h50, depending on the solution chosen.
