New characterizations of the solutions to overdetermined systems of linear equations are given. The first is a polyhedral characterization of the solution set in terms of a special sign vector using a simple property of the solutions. The second characterization is based on a smooth approximation of the function using a ‘Huber’ function. This allows a description of the solution set of the problem from any solution to the approximating problem for sufficiently small positive values of an approximation parameter. A sign approximation property of the Huber problem is also considered and a characterization of this property is given.
