Skip to main content
1 of 6
Hao Chen
  • 2.6k
  • 19
  • 29

This answer benefited from a discussion with I. Izmestiev.

Consider a $3$-dimensional polytope $P$. The indegrees of the vertices can only be $0$, $1$, $2$ or $3$. The lowest vertex has indegree $0$, the highest vertex has indegree $3$, and the other vertices have indegree $1$ or $2$. For an index-increasing realization of $P$, there is a hyperplane that separates the indegree-$1$ vertices and the indegree-$2$ vertices.

Moreover, every face of $P$ contains a vertex of indegree $1$ and a vertex of indegree $2$. Therefore, the existence of the required realization means that there must be a hyperplane that cuts all the faces.

From this point of view, it becomes much clearer that index-increasing realization is very unlikely to exist.

Hao Chen
  • 2.6k
  • 19
  • 29