Hyperspace of max-plus convex compact sets

The hyperspace mpcc(? ⁿ ) of max-plus convex compact subsets of ? ⁿ , n ≥ 2, 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 mpcc(? ⁿ ) → mpcc(? ^m ), n ≥ m, is open. Moreover, it is proved that the hyperspace mpcc( $ I^{omega _1 } $ ) of the Tikhonov [Tychonoff] cube $ I^{omega _1 } $ is homeomorphic to $ I^{omega _1 } $ .