Neighborly polytope

In geometry and polyhedral combinatorics, a k-neighborly polytope is a convex polytope in which every set of k or fewer vertices forms a face. For instance, a 2-neighborly polytope is a polytope in which every pair of vertices is connected by an edge, forming a complete graph. 2-neighborly polytopes with more than four vertices may exist only in spaces of four or more dimensions, and in general a k-neighborly polytope (other than a simplex) requires a dimension of 2k or more. A d-simplex is d-neighborly. A polytope is said to be neighborly, without specifying k, if it is k-neighborly for ${\displaystyle k=\lfloor d/2\rfloor }$. If we exclude simplices, this is the maximum possible k: in fact, every polytope that is k-neighborly for some ${\displaystyle k\geq 1+\lfloor d/2\rfloor }$ is a simplex.

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