On a problem of Namioka on norm-attaining functionals

Isaac Namioka conjectured that every nonreflexive Banach space can be renormed is such a way that, in the new norm, the set of norm attaining functionals has an empty interior in the norm topology. We prove the rightness of this conjecture for spaces containing an isomorphic copy of ℓ₁. As a consequence, we prove also that the same result holds for a wide class of Banach spaces containing, for example, the weakly compactly generated ones.