## Geometry of quotient spaces and proximinality

### Volume 82 / 2003

#### Abstract

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 ${\mit \Phi }$ does not satisfy the $\delta _2$-condition, then $h_{{\mit \Phi }}^0 $ is not proximinal in $l_{{\mit \Phi }}^0$ and the quotient space $l_{{\mit \Phi }}^0/ h_{{\mit \Phi }}^0$ is not rotund (even if $l_{{\mit \Phi }}^0$ 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 uniform Kadec–Klee property. It is noted that the quotient space $X/M$ with $X$ and $M$ as above is weakly nearly uniformly convex whenever $X$ is weakly nearly uniformly convex. Criteria for weakly nearly uniform convexity of Orlicz sequence spaces equipped with the Orlicz norm are given.