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Abstract

Let $E$ be a Banach space and let ${\cal B}_1(B_{E^*})$ and ${\mathfrak A}_1(B_{E^*})$ denote the space of all Baire-one and affine Baire-one functions on the dual unit ball $B_{E^*}$, respectively. We show that there exists a separable $L_1$-predual $E$ such that there is no quantitative relation between $\mathop{\rm dist}(f,{\cal B}_1(B_{E^*}))$ and $\mathop{\rm dist}(f,{\mathfrak A}_1(B_{E^*}))$, where $f$ is an affine function on $B_{E^*}$. If the Banach space $E$ satisfies some additional assumption, we prove the existence of some such dependence.