  Higher Auslander algebras were introduced by Iyama generalizing classical
concepts from representation theory of finite dimensional algebras. Recently
these higher analogues of classical representation theory have been
increasingly studied. Cyclic polytopes are classical objects of study in convex
geometry. In particular, their triangulations have been studied with a view
towards generalizing the rich combinatorial structure of triangulations of
polygons. In this paper, we demonstrate a connection between these two
seemingly unrelated subjects.
  We study triangulations of even-dimensional cyclic polytopes and tilting
modules for higher Auslander algebras of linearly oriented type A which are
summands of the cluster tilting module. We show that such tilting modules
correspond bijectively to triangulations. Moreover mutations of tilting modules
correspond to bistellar flips of triangulations.
  For any d-representation finite algebra we introduce a certain d-dimensional
cluster category and study its cluster tilting objects. For higher Auslander
algebras of linearly oriented type A we obtain a similar correspondence between
cluster tilting objects and triangulations of a certain cyclic polytope.
  Finally we study certain functions on generalized laminations in cyclic
polytopes, and show that they satisfy analogues of tropical cluster exchange
relations. Moreover we observe that the terms of these exchange relations are
closely related to the terms occuring in the mutation of cluster tilting
objects.
