Testwiki:Reference desk/Archives/Mathematics/2024 January 16

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January 16

Convex polytopes where any two vertices are connected by an edge.

This is true of simplices, are there others? I believe there are, and if so is there a name for them? We have an article on simplicial polytopes, which seems related. There is a theorem that a simple polytope is completely determined by its 1-skeleton, see Simple polytope#Unique reconstruction, and in this case that's a complete graph. But all this proves is that such a polytope, other than a simplex, is not simple. RDBury (talk) 13:59, 16 January 2024 (UTC)

Such a polytope with $k+1$ vertices should be fundamentally identical to the projection of an $k$-simplex to whichever dimension you're working in. Also, if I'm not mistaken, if we take the "volume" occupied by such a projection to be the projection of the volume occupied by the simplex in $k$ dimensions, then regardless of dimension, the resulting polytope should be convex. GalacticShoe (talk) 16:20, 16 January 2024 (UTC)

Good point. I don't often play with non-convex polytopes and tend to conflate "polytope" with "convex polytope", so I was being careful to specify "convex". In fact, any convex polytope which is the convex hull of its vertices is the projection of a simplex of the same or higher dimension. For the purposes of the question I'm thinking of "proper polytopes", bounded and determined both as the intersection of a finite set of half-planes and as the convex hull of a finite set of points. --RDBury (talk) 17:35, 16 January 2024 (UTC)

The cyclic polytopes have this property (and much stronger properties) in dimension 4 and above. In 2-D the question is trivial, while in 3-D there is one (known) non-convex example, the Császár polyhedron (and no convex examples other than the tetrahedron). --JBL (talk) 21:01, 16 January 2024 (UTC)

Good examples. Interesting that the 3-D one is toroidal. For convex polyhedra the Euler V-E+F=2 relation rules them out except for the tetrahedron. RDBury (talk) 14:18, 17 January 2024 (UTC)

It seems that the word for the concept you're interested in is 2-neighborly polytope. --JBL (talk) 21:04, 16 January 2024 (UTC)

Yes, that was the term I was looking for. I'll peruse the article and do a Google search. Thanks. RDBury (talk) 14:27, 17 January 2024 (UTC)

You're welcome :). --JBL (talk) 18:00, 17 January 2024 (UTC)