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Too long for a comment: I think there is a reason, why this is algorithmically solvable. Given three side lengths $a,b$ and $c$, and a fixed number $k\in\mathbb{N}$, writing down all the (in)equali …

The answer to your question is: no Its also easy to give a concrete counter-example: Take the cube with vertices $$ \left(\frac{273}{340},\,\frac{79}{68},\,\frac{13}{20}\right) , \left(\frac{407 …

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 …

I hope you can find a partial answer here. At least for $n=d+2$.

A (trivial) necessary condition is that the diagonal of the inner one is not longer than the diagonal of the outer one. So if $(a,b,c)$ is supposed to fit in $(x,y,z)$, then we should have $$a^2+b^2+ …

Yashar Mermarian writes here that the answer is yes. And the argument he gives is pretty much the same as the one you already started. Taking $n=k$ and $\epsilon=\pi/2$ Gromov's waist inequality give …

When studying this problem in the general case, i.e. $m$-dimensional cube inside $n$-dimensional cube for $m<n$, it might make sense to look at small examples to gain some intuition. From the small ex …

Since I expect the answer for all combinatorial types of polytopes to be somewhat obscure, let me give an answer to the more simple question of what happens for the tetrahedron. We denote the six edge …

It seems in the meantime, there are new results: Bai, Zai-Qiao; Finch, Steven R. Precise calculation of Hausdorff dimension of Apollonian gasket. Fractals 26 (2018), no. 4, 9 pp. claims a better appro …

Looking at this old question again, I'm now fairly convinced that the easiest route of solving this problem is using similar ideas to the one suggested by David E Speyer in a comment, namely basically …

Here is an idea. Consider the following parameterization, which is supposed to cover the configuration space in question. $$\mathcal{C}_7:=\left\{\pmatrix{x_k\\y_x\\z_k},\pmatrix{a_k\\b_k\\c_k}_{1\le …

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 …

Using global nonlinear optimization one can obtain a configuration of $19$ spheres, that touch at least one of the central unit spheres and have almost no overlap. In fact, if one takes their radii to …