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For the tetrahedron, this calculation comes up in the Rogers sphere packing bound. The answer is $\sqrt{18}(\cos^{-1}(1/3) - \pi/3)$, which is roughly $0.7796$. For the octahedron, one can calculate …

The question is slightly ambiguous, since it doesn't specify how large the linear program can be or how much preprocessing can be devoted to producing it. However, if everything is required to run in …

It’s almost certainly true, and provable, that $\alpha=\sqrt{6}$, although I haven’t worked out the details rigorously. Kabatiansky and Levenshtein give an exact upper bound (not just an asymptotic ex …

Nope, this is false in three dimensions (where the body-centered cubic beats the face-centered cubic for covering) and eight dimensions (where $E_8$ is not even locally optimal). The Leech lattice is …

To understand the cases of spherical or hyperbolic geometry, it is helpful to think in terms of spherical codes. A Euclidean kissing configuration is equivalent to an arrangement of points on a spher …

Pietro's version of this question is answered in a paper by Oleg Musin and Alexey Tarasov (to appear in Discrete & Computational Geometry, http://dx.doi.org/10.1007/s00454-011-9392-2, http://arxiv.org …

There’s definitely a lot of potential for finding great packings using computers. I don’t believe the known sphere packings up through 24 dimensions are all optimal, and a clever heuristic algorithm c …

In three dimensions you don’t need to go beyond lattices to see the failure of universal optimality. When the potential function is sufficiently steep (e.g., a narrow Gaussian), the face-centered cubi …

I haven't proved anything, just done some numerical experiments, but I do not think there is always a two-parameter family of maximum-surface-area tetrahedra incribed in an ellipsoid (although you do …

There exist coverings such that each point is covered at most $400 d \log d$ times, and you can improve this bound a little if you look at the covering density, i.e., the average number of times each …

There are two things you need to understand. The first is how to prove sphere packing bounds via harmonic analysis ("linear programming bounds"). My lecture notes from PCMI 2014 give an exposition o …

As I see it, the key intuition is passing from the equator orthogonal to a single vector to looking at a whole orthonormal basis. Suppose we pick a random unit vector $(x_1,\dots,x_n)$. What we want …

It's known; for example, my coauthors and I used this characterization in http://arxiv.org/abs/math.MG/0611451. However, we certainly weren't the first, and I don't know the earliest reference. I th …

An $8$-vertex polyhedron can achieve an isoperimetric ratio of $A^3/V^2 = 159.3243297053\dots$, and based on some quick experiments I'm pretty confident this is optimal (although I wouldn't be shocked …