The AR-Property of the spaces of closed convex sets
Volume 106 / 2006
Colloquium Mathematicum 106 (2006), 15-24
MSC: 54B20, 54C55, 46A55.
DOI: 10.4064/cm106-1-2
Abstract
Let $\mathop{\rm Conv}_{\rm H}(X)$, $\mathop{\rm Conv}_{\rm AW}(X)$ and $\mathop{\rm Conv}_{\rm W}(X)$ be the spaces of all non-empty closed convex sets in a normed linear space $X$ admitting the Hausdorff metric topology, the Attouch–Wets topology and the Wijsman topology, respectively. We show that every component of $\mathop{\rm Conv}_{\rm H}(X)$ and the space $\mathop{\rm Conv}_{\rm AW}(X)$ are AR. In case $X$ is separable, $\mathop{\rm Conv}_{\rm W}(X)$ is locally path-connected.
