On the openness of the idempotent barycenter map

Abstract

We show that the openness of the idempotent barycenter map is equivalent to the openness of the map of Max-Plus convex combination. As corollary we obtain that the idempotent barycenter map is open for the spaces of idempotent measures.2010 Mathematics Subject Classification. 52A30; 54C10; 28A33. Key words and phrases. open map; idempotent (Maslov) measure; idempotent barycenter map.

Bundle of idempotent measures

“…In this section we investigate the map β IX : I 2 X → IX for a metrizable compactum X. It was proved in [14] that β IX : I 2 X → IX is open for each compactum X.…”

“…The set of extremal points of a max-plus convex compactum X we denote by E(X). It is proved in [14] that the set E(X) is closed if the idempotent barycenter map is open.…”

“…compact convex sets for which the barycentre map is open) is again barycentrically open. It was shown in [14] that analogous statements for idempotent measures are false.…”

“…]) are absolute retracts. Since the map β [0,1] is open [14], β [0,1] is soft (Corollary 4.2 in [17]). The the maps b 0 and b 1 are soft too being restrictions of β [0,1] to complete preimages.…”