Taylor expansion of best lp-approximations about p=1

In this paper, we consider the problem of best approximation in l(p)(n), 1 less than or equal to p less than or equal to infinity. If h(p), 1 less than or equal to p < infinity, denotes the best l(p)-approximation of the element h is an element of R-n from a proper affine subspace K of R-n, h is not an element of K, then lim(p -> 1) h(p) = h(1)*, where h(1)*, is a best l(1)-approximation of h from K, the so-called natural best l(1)-approximation. We prove that, for every r is an element of N, the best l(p)-approximations have a Taylor expansion of order r of the form [GRAPHICS] for some alpha(l) is an element of R-n, 1 <= l <= r, and gamma((r))(p) is an element of R-n with parallel togamma(p)((r))parallel to = O((p-1)(r+1)) (C) 2003 Elsevier Ltd. All rights reserved.