  The Dantzig selector (Candes and Tao, 2007) is a popular l1-regularization
method for variable selection and estimation in linear regression. We present a
very weak geometric condition on the observed predictors which is related to
parallelism and, when satisfied, ensures the uniqueness of Dantzig selector
estimators. The condition holds with probability 1, if the predictors are drawn
from a continuous distribution. We discuss the necessity of this condition for
uniqueness and also provide a closely related condition which ensures
uniqueness of lasso estimators (Tibshirani, 1996). Large sample asymptotics for
the Dantzig selector, i.e. almost sure convergence and the asymptotic
distribution, follow directly from our uniqueness results and a continuity
argument. The limiting distribution of the Dantzig selector is generally
non-normal. Though our asymptotic results require that the number of predictors
is fixed (similar to (Knight and Fu, 2000)), our uniqueness results are valid
for an arbitrary number of predictors and observations.
