It is proved that if X is a rotund Banach space and M is a closed and proximinal subspace of X, then the quotient space X/M is also rotund. It is also shown that if Φ does not satisfy the δ₂-condition, then $h{⁰}_{\Phi }$ is not proximinal in $l{⁰}_{\Phi }$ and the quotient space $l{⁰}_{\Phi }/h{⁰}_{\Phi }$ is not rotund (even if $l{⁰}_{\Phi }$ is rotund). Weakly nearly uniform convexity and weakly uniform Kadec-Klee property are introduced and it is proved that a Banach space X is weakly nearly uniformly convex if and only if it is reflexive and it has the weakly...