The One and Half Ball Property in Spaces of Vector-Valued Functions

We exhibit new classes of Banach spaces that have the strong-$1\frac{1}{2}$-ball property and the $1\frac{1}{2}$-ball property by considering direct-sums of Banach spaces. We introduce the notion of sectional strong-$1\frac{1}{2}$-ball property and show that in $c_0$-direct sum of reflexive spaces, proximinal and factor reflexive spaces with the sectional strong-$1\frac{i}{2}$-ball property have the strong-$1\frac{1}{2}$-ball property. We give examples of proximinal hyperplanes in $c_0$ that fail the $1\frac{1}{2}$-ball property and show that this property is in general, not preserved under finite intersections or sums. We show that the range of a bi-contractive projection in $\ell^{\infty}$ has the strong-$1\frac{1}{2}$-ball property. For a separable subspace $Y \subset X$ with the strong-$1\frac{1}{2}$-ball property and for any positive, $\sigma$-finite, non-atomic measure space $(\Omega, {\mathcal A}, \mu)$, we show that $L^1(\mu,Y)$ has the strong-$1\frac{1}{2}$-ball property in $L^1(\mu,X)$. We show that for any compact set $\Omega$ and $Y \subset X$ with the $1\frac{1}{2}$-ball property, $C(\Omega,Y)$ has the $1\frac{1}{2}$-ball property in $C(\Omega,X)$.