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If X is a Banach space and C a convex subset of X*, we investigate whether the distance from to C is M-controlled by the distance d̂(K,C) (that is, if for some 1 ≤ M < ∞), when K is any weak*-compact subset of X*. We prove, for example, that: (i) C has 3-control if C contains no copy of the basis of ℓ₁(c); (ii) C has 1-control when C ⊂ Y ⊂ X* and Y is a subspace with weak*-angelic closed dual unit ball B(Y*); (iii) if C is a convex subset of X and X is considered canonically embedded into...
If X is a Banach space and C ⊂ X a convex subset, for x** ∈ X** and A ⊂ X** let d(x**,C) = inf||x**-x||: x ∈ C be the distance from x** to C and d̂(A,C) = supd(a,C): a ∈ A. Among other things, we prove that if X is an order-continuous Banach lattice and K is a w*-compact subset of X** we have: (i) and, if K ∩ X is w*-dense in K, then ; (ii) if X fails to have a copy of ℓ₁(ℵ₁), then ; (iii) if X has a 1-symmetric basis, then .
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