Hyperspace of Max-Plus Convex Compact Sets

The hyperspace $\operatorname{mpcc}(\mathbb R^n)$ of max-plus convex compact subsets of $\mathbb R^n$, $n\ge2$, is considered. The main result is as follows: this hyperspace is a contractible manifold modeled on the Hilbert cube $Q$. It is also shown that the projection mapping $\operatorname{mpcc}(\mathbb R^n) \to\operatorname{mpcc}(\mathbb R^m)$, $n\ge m$, is open. Moreover, it is proved that the hyperspace $\operatorname{mpcc}(I^{\omega_1})$ of the Tikhonov [Tychonoff] cube $I^{\omega_1}$ is homeomorphic to $I^{\omega_1}$.