Given $K$ points in the $N$-simplex, when is there a unique $M$-polytope that contains the $K$ points?
For example, in the below image $K=13, N= 2, M=3$. Clearly the polytope (the inner triangle) can be moved around while still containing those points, so it is nonunique.
Here are some simple cases I can think of:
- $N=K=M$, and the points are the vertices of the simplex.
- $K,N>M$, and $M$ of the points are on the faces of the simplex.
What is a general criterion?
Related question: when does no such polytope exist? This can happen if $N,K>M$ and there are more than $K$ distinct points on the faces of the simplex. Is this the only case?