# Is every planar point set the projection of vertices of a neighborly 4-polytope?

More exactly, written in coordinates, I'm curious to know if for every point set $(x_i,y_i)$ there are $(x_i,y_i,z_i,w_i)$ that are vertices of a neighborly polytope.

This problem comes from a simple observation: every planar point set is projection of a convex 3-polytope. One can lift $(x,y)\mapsto (x,y,x^2+y^2)$ as the inverse of projection.

There is the oriented matroid version of such problem, can topes of every acyclic rank-3 uniform oriented matroid realized as a subset of topes of rank-5 neighborly oriented matroid? (I'm also not very sure the whether taking subset and projection is always the same, this is the strong map conjecture) I'm looking for contour examples for both problem.

• The parabolic lift map "works" in arbitrary dimensions, which does not directly answer your question, but... Commented Feb 7, 2015 at 2:44
• @JosephO'Rourke I've tried to construct a "neighborly" surface embedded in R^4, which should have properties that the line connecting two points in the surface should be intersection of tangent planes of two point, and I believe the surface (x,y,x^2+y^2,x^2+y^2+(x^2+y^2)^2) do not have such properties. Commented Feb 7, 2015 at 2:50
• OK, @Belanov, I see that I haven't fully considered your neighborly requirement. My oversight. Commented Feb 7, 2015 at 2:57