Properties of Chebyshev Generalized Rational Fractions in L1

It is shown that, under a natural constraint, a set of generalized rational fractions in an atomless L-1-space is a Chebyshev set with continuous metric projection only if this set is convex. Hence this set is not a uniqueness set in L-1, and therefore, some x is an element of L(1 )has at least two nearest points in this set. As a result, it is shown that the set of classical algebraic fractions R-n,R-m (consisting of ratios of algebraic polynomials of degree <= n, <= m, respectively) is not a Chebyshev set in L-1[a, b], and therefore, there exists a function x is an element of L-1[a, b] with at least two nearest points in R-n,R-m. This result solves one long-standing problem in rational approximation.