Norm-attaining functionals need not contain 2-dimensional subspaces

G. Godefroy asked whether, on any Banach space, the set of norm-attaining functionals contains a 2-dimensional linear subspace. We prove that a construction due to C.J. Read provides an example of a space which does not have this property. Read found an equivalent norm ⦀⋅⦀ on c0 such that (c0,⦀⋅⦀) contains no proximinal subspaces of codimension 2. Our result is obtained through a study of the relation between the following two sentences, in which X is a Banach space and Y⊂X is a closed subspace: (A) Y is proximinal in X, and (B) Y⊥ consists of norm-attaining functionals. We prove that these are equivalent if X is the space (c0,⦀⋅⦀), and our main theorem then follows as a corollary to Read's result.