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Maybe a term you are looking for is "sparse solutions of underdetermined linear systems". At least this would maybe be a good name for your set $V$. It can be viewed as solution set of the following o …

Too long for a comment: The $f$-vector of the polytopes in question appear to be: P_1 (1, 2, 1) P_2 (1, 4, 4, 1) P_3 (1, 5, 8, 5, 1) P_4 (1, 9, 18, 15, 6, 1) P_5 (1, 10, 27, 33, 21, 7, 1) P_6 (1, 16, …

If I understand correctly, your polytopes are a subset of the polytopes where every inequality has at most two variables with non-zero coefficient. Those are studied in On the Complexity of Polytope …

I can think of a few purely combinatorial criteria, that allow to deduce realizability as a polytope. All d-polytope with at most d+2 vertices is realizable Stacked polytopes. (It can be easily comb …

Allow me look at one aspect, or special case, of your question, namely "finding the largest regular 3-dimensional tetrahedron inscribed in a d-dimensional unit cube". I. $4$-cube I can find the follow …

I don't know any reference for this, and I don't know if this should be a "classical result", but let me give a lower bound, which might even be tight. Let's denote the base of the cap by $BS$. It is …

Too long for a comment: Stringham gave a talk about the content of his thesis here in the Seminar of Felix Klein in Göttingen on Monday, 1880/11/29, you can look at the scans here: Ueber reguläre Körp …

You can find Frank Lutz's lists of simplicial spheres (and other manifolds) here: Let me shamelessly self-advertise my list of simplicial 4-polytopes with up to $10$ vertices and various families of …

Here are four combinatorial types of $4$-polytopes with $6$ vertices: a. The pyramid over the pyramid over the square. b. The pyramid over the bipyramid over the triangle c. The bipyramid over the …

Alathea Jensen defines "self-polar": Self-polar polytopes are convex polytopes that are equal to an orthogonal transformation of their polar sets. and writes some interesting things about self-p …

I guess it would be difficult to prove that the answero your question is "no", since proving that "no a priori reason exists" might be hard. More modestly, I can say that I don't really know a good r …

Edit: a preprint concerning this problem can now be found on the arXiv: http://arxiv.org/abs/1407.0683 Let me give an exhaustive answer. Croft closes his paper with a list of the unsolved cases: Here …

I used sage to make a 3d animation of all 261 unfoldings. Here is a screenshot of the first few: The file cube-unfoldings.txt contains all the unfoldings, each line contains a list of 8 points. …

Answer to Q1: All of the 261.  I looked at this question because of a video of Matt Parker and wrote an algorithm to find solutions. See here for an example of how a solution would look like. I dumped …

While it is clear that the regular dodecahedron $D$ cannot be realized with all integer coordinates, it is easy to find a polytope, which is combinatorially equivalent (face lattice isomorphic) to $D$ …