By E. W. Cheney
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A subspace of finite co-dimension in C(S) has property EK if and only if it is proximinal. In contrast to this, Zamyatin [1973] has proved: THEOREM. In order that a subspace U of finite co-dimension in C(S) have property EO it is necessary and sufficient that each element of U ⊥ have finite support. From these two theorems it is clear that some hyperplanes in C(S) have property EK but not property EO. Further references on the problem of Chebyshev centers are Amir and Ziegler [1980-a, 1980-b, 1981], Blatt [1973], Bosznay [1978], Brondsted [1976], Carroll [1972], Chui, Rahman, Sahney and Smith [1978], Diaz and McLaughlin [1969-a], Dierieck [1976], Dunham [1967], Franchetti [1977], Gillotte and McLaughlin [1976], Hall [1976], Holland, Sahney, and Tzimbalario [1976], Kadets and Zamyatin [1968], Laurent and Tuan [1970], Lin [1974], Mach [1979-a, 1979-b, 1982], Milman [1977], Sahney and Singh [1982], and Owens [1983].
The space L1 (S, X) consists of all Bochner integrable maps from S to X. The elementary theory of these spaces is developed in Chapter 10 of Light and Cheney [1985]. The book by Diestel and Uhl [1977] gives more complete information. THEOREM. Let S and T be finite measure spaces. Let G and H be finitedimensional subspaces in L1 (S) and L1 (T ), respectively. Then G ⊗ L1 (T ) + L1 (S) ⊗ H is proximinal in L1 (S × T ). THEOREM. Let S and T be σ-finite measure spaces. Let G and H be finitedimensional subspaces in L∞ (S) and L∞ (T ), respectively.
A minimal projection of Lp (S × T ) onto Lp (S) + Lp (T ) can be constructed as follows. Define P on Lp (S) and Q on Lp (T ) by Px = x(s)ds, Qy = y(t)dt. Then (P ⊗ I) ⊕ (I ⊗ Q) is the minimal projection sought. Here we have assumed for simplicity that S and T have measure 1. This result has been proved by Halton and Light [1985b]. An interesting open problem here is whether spaces of the type W = X ⊗H +G⊗Y always possess minimal projections from X ⊗α Y, it being assumed that G and H are finite-dimensional subspaces.