# On the openness of the idempotent barycenter map

## TL;DR

This paper investigates the conditions under which the idempotent barycenter map is open, establishing its equivalence to the openness of the Max-Plus convex combination map and applying this to spaces of idempotent measures.

## Contribution

It demonstrates the equivalence of openness between the idempotent barycenter map and the Max-Plus convex combination map, providing new insights into their properties.

## Key findings

- Idempotent barycenter map is open iff Max-Plus convex combination map is open.
- The map is open for spaces of idempotent measures.
- Establishes a key equivalence in the theory of idempotent measures.

## 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.