ABSTRACT. In [1] we studied a certain class of “one-parameter” Chebyshev sets in normed linear spaces (call EU-regular Chebyshev sets), and we obtained necessary and sufficient conditions for these sets to the suns. We showed that if T is a compact metric space, then C(T) always contains an EU-regular Chebyshev set that is not a sun. (Dunham [4] had given the first example of a Chebyshev set in any normed linear space that was not a sun. His example was in C[0, 1].)

In this paper it is our purpose to show that the situation in L_{1} space is quite different. Indeed, every EU-regular Chebyshev set in an L_{1} space is a sun.