Absolutely Chebyshev subspaces

Abstract

Let's say that a closed subspace $M$ of a Banach space $X$ is absolutely Chebyshev if it is Chebyshev and, for each $x \in X, \| x \|$ can be expressed as a function of only $d(x,M)$ and $\| P_m(x)\|$. A typical example is a dosed subspace of a Hilbert space. Absolutely Chebyshev subspaces are, modulo renorming, the same as semi-$L$-summands. We show that any real Banach space can be absolutely Chebyshev in some larger space, with a nonlinear metric projection. Dually, it follows that if $M$ has the 2-ball property but not the 3-ball property in $X$, no restriction exists on the quotient space $X/M$. It is not known whether such examples can be found in complex Banach spaces.