This is an announcement for the paper “On proximinality of subspaces of finite codimension of Banach spaces and the lineability of the set of norm-attaining functionals” by Miguel Martinhttps://arxiv.org/search?searchtype=author&query=Martin%2C+M.
Abstract: We show that for every $1<n<\infty$, there exits a Banach space $X_n$ containing proximinal subspaces of codimension $n$ but no proximinal finite codimensional subspaces of higher codimension. Moreover, the set of norm-attaining functionals of $X_n$ contains $n$-dimensional subspaces, but no subspace of higher dimension. This gives a $n$-by-$n$ version of the solutions given by Read and Rmoutil to problems of Singer and Godefroy. Actually, the space $X_n$ can be found with strictly convex dual and bidual, and such that the slices of its unit ball have diameter as close to two as desired. We also deal with the existence of strongly proximinal subspaces of finite codimension, showing that for every $1<n<\infty$ and $1\leq k <n$, there is a Banach space $X_{n,k}$ containing proximinal subspaces of finite codimension up to $n$ but not higher, and containing strongly proximinal subspaces of finite codimension up to $k$ but not higher.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1805.05979