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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 …

Not all simple polytopes are incribable, e.g. the dual of the cyclic polytope $C_4(8)$ is simple and not inscribable, as shown recently in Combinatorial Inscribability Obstructions for Higher-Dimensio …

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 $D$ …

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 …

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 …

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. …

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 …

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 …

Let me answer at least some of your questions. I will only talk about your first definition of the cells, since these are somewhat nicer, as Igor Rivin pointed out. You consider the function $f(w_1)= …

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 …

The state-of-the-art methods for proving non-polytopality differ from those of proving polytopality of a simplicial sphere $S$. Let me give you an overview: If $S$ is non-polytopal, then one way of …

Looking at the last paragraph of your question, it might seem reasonable to reformulate the question as: can we always select weights $\mathbf{w}$, such that the values $Q(V_i(\mathbf{w}))$ satisfy pr …

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 …

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 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 …