Taylor expansion of best lp-approximations about p = 1

In this paper, we consider the problem of best approximation in e,(n), 1 5 p 5 co. If h,, 1 5 p < co, denotes the best &-approximation of the element h E W " from a proper affine subspace K of W " , h @ K, then limp,1 h p-h* where h; is a best er-approximation of h from K, the i, so-called natural best er-approximation. We prove that, for every T E A, the best &approximations have a Taylor expansion of order T of the form for some crl E Wn, 1 5 1 5 T, and TV' E W " with I[rg' 11 = O((p-l)r+l). @ 2003 Elsevier Ltd. All rights reserved.