  We study geometric properties of the Banach space $\mathcal{R}$ constructed
recently by C.~Read (arXiv 1307.7958) which does not contain proximinal
subspaces of finite codimension greater than or equal to two. Concretely, we
show that the bidual of $\mathcal{R}$ is strictly convex, that the norm of the
dual of $\mathcal{R}$ is rough, and that $\mathcal{R}$ is weakly locally
uniformly rotund (but it is not locally uniformly rotund). Apart of the own
interest of the results, they provide a simplification of the proof by
M.~Rmoutil (J.\ Funct.\ Anal.\ 272 (2017), 918--928) that the set of
norm-attaining functionals over $\mathcal{R}$ does not contain any linear
subspace of dimension greater than or equal to two. Note that if a Banach space
$X$ contains proximinal subspaces of finite codimension at least two, then the
set of norm-attaining functionals over $X$ contain two-dimensional linear
subspaces of $X^*$. Our results also provides positive answer to the questions
of whether the dual of $\mathcal{R}$ is smooth and of whether $\mathcal{R}$ is
weakly locally uniformly rotund (J.\ Funct.\ Anal.\ 272 (2017), 918--928).
Finally, we present a renorming of Read's space which is smooth, whose dual is
smooth, and such that its set of norm-attaining functionals does not contain
any linear subspace of dimension greater than or equal to two, so the renormed
space does not contain proximinal subspaces of finite codimension greater than
or equal to two.
