Topological classification of the hyperspaces of polyhedral convex sets in normed spaces

We prove that, for a normed space $X$ of dimension $dim(X)ge2$ the space $PConv_H(X)$ of non-empty polyhedral convexsubsets of $X$ endowed with the Hausdorff metric ishomeomorphic to the topological sum ${0}oplus |X^*| imes(IRoplus (IR imesarIR_+)oplus ssigma)$, where thecardinal $|X^*|$ is endowed with the discrete topology.