Vertices of balls: orders, codes, and tropical geometry

Balls in the tropical metric are a particular example of polytropes, namely tropical polytopes that are at the same time convex in the usual sense. Moreover, tropical balls can also be seen as one-apartment slices of balls in Bruhat-Tits buildings. So balls in this context have vertices and we will see how these vertices play an important role in applications to representation theory and algebraic coding theory. I will be reporting on joint work with Yassine El Maazouz, Marvin Hahn, Gabriele Nebe, and Bernd Sturmfels.