We show that linear programs (LPs) admit regularizations that either contract the original (primal) solution set or leave it unchanged. Any regularization function that is convex and has compact level sets is allowed; differentiability is not required. This is an extension of the result first described by Mangasarian and Meyer (SIAM J. Control Optim., 17(6), pp. 745-752, 1979). We show that there always exist positive values of the regularization parameter such that a solution of the regularized problem simultaneously minimizes the original LP and minimizes the regularization function over the original solution set. We illustrate the main result using the nondifferentiable one-norm regularization function on a set of degenerate LPs. Numerical results demonstrate how such an approach yields sparse solutions from the application of an interior-point method.