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

We prove that, for a normed space $X$ of dimension $dim(X)ge 2$ the space $mathrm{PConv}_H(X)$ of non-empty polyhedral convex subsets of $X$ endowed with the Hausdorff metric is homeomorphic to the topological sum ${0}oplus |X^*|times (mathbb Roplus (mathbb Rtimesbar{mathbb{R}}_+)oplus l_2^f)$, where the cardinal $|X^*|$ is endowed with the discrete topology.