Bazylevych L. Hyperspaces of convex sets

The thesis is devoted to the study of topology of hyperspaces of classes of convex bodies and their maps. Convexity theories in the idempotent mathematics are investigated and it is proved, in particular, that the hyperspace of max-plus and max-min convex compacta in the euclidean spaces is a contractible Q-manifold. It is proved that the hyperspace of max-plus convex subsets in Tychonov cube of weight aleph-1 is homeomorphic to this cube and necessary and sufficient conditions are established in order that the hyperspace of convex compacta in the Tychonov cube (respectively the cone over Tychonov cube) are homeomorphic to the Tychonov cube (respectively the cone over Tychonov cube).