On best uniform approximation by low-rank matrices

We study the problem of best approximation, in the elementwise maximum norm, of a given matrix by another matrix of lower rank. We generalize a recent result by Pinkus that describes the best approximation error in a class of low-rank approximation problems and give an elementary proof for it. Based on this result, we describe the best approximation error and the error matrix in the case of approximation by a matrix of rank one less than the original one. For the case of approximation by matrices with arbitrary rank, we give lower and upper bounds for the best approximation error in terms of certain submatrices of maximal volume. We illustrate our results using 2×2 matrices as examples, for which we also give a simple closed form of the best approximation error.