A motivation of this paper is to explore the idea that the L1 norm might produce the kind of solutions that Daniel Busch would like to see. I'm far from certain of this. The behavior of the L1 norm is well understood in overdetermined formulations. There L1 is valuable because it rejects outliers (large residuals). The behavior of the L1 norm in underdetermined problems is not well understood nor has it been widely observed.
One reason the behavior of the L1 norm is not widely observed in underdetermined problems is that we do not have fast and reliable computational methods for problems of high dimensionality. For example, in Busch's example the unknowns are a cartesian mesh of all possible altitudes where just four altitudes are constrained. (Actually, we probably also need to specify the behavior at infinite distance.)
To see if L1 might help us solve problems like the one posed by Busch, and to help us guess whether we should invest resources in L1 solvers, I review here various examples of lesser scope.