
Journal of Convex Analysis 20 (2013), No. 1, 013023 Copyright Heldermann Verlag 2013 The One and Half Ball Property in Spaces of VectorValued Functions T. S. S. R. K. Rao Indian Statistical Institute, R. V. College P.O., Bangalore 560059, India tss@isibang.ac.in [Abstractpdf] 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 directsums 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 bicontractive 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, nonatomic 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 $C(\Omega,X)$. Keywords: One and half ball property, spaces of vectorvalued functions. MSC: 41A65,46B20; 41A50 [ Fulltextpdf (142 KB)] for subscribers only. 