  In this paper, we study the solution uniqueness of an individual feasible
vector of a class of convex optimization problems involving convex piecewise
affine functions and subject to general polyhedral constraints. This class of
problems incorporates many important polyhedral constrained $\ell_1$ recovery
problems arising from sparse optimization, such as basis pursuit, LASSO, and
basis pursuit denoising, as well as polyhedral gauge recovery. By leveraging
the max-formulation of convex piecewise affine functions and convex analysis
tools, we develop dual variables based necessary and sufficient uniqueness
conditions via simple and yet unifying approaches; these conditions are applied
to a wide range of $\ell_1$ minimization problems under possible polyhedral
constraints. An effective linear program based scheme is proposed to verify
solution uniqueness conditions. The results obtained in this paper not only
recover the known solution uniqueness conditions in the literature by removing
restrictive assumptions but also yield new uniqueness conditions for much
broader constrained $\ell_1$-minimization problems.
