I'm curious if in your experiments included knots (physical knots, mathematically really unknots) designed to be loaded on the free ends. The two that come to mind are the alpine butterfly and the sheepshank. A figure-8 on a bight is also worth trying, though I can see that rolling with frictionless rope.

related: mathoverflow.net/questions/118481/… You're asking about the measure of the intersection of the positive-semidefinite cone with the trace=1 hyperplane, rather than the unit sphere, but maybe the linked question and answer could be helpful.

@DanielAsimov I think "uniformly recurrent packings", as described by Greg Kuperberg in "Notions of denseness" (dx.doi.org/10.2140/gt.2000.4.277) is sort of the idea you're reaching for with "tails of packings", and it overcomes the problem of asymptotically-zero-density deformations that Peter points out. In fact, in that paper, Greg indicates (if I understand correctly) that it's likely that the only uniformly recurrent ball packings of maximum density are Barlow packings.

Also note that, e.g., if you remove one ball from the fcc packing, this doesn't affect the asymptotic density, so could arguably be said to be as efficient. To formulate your question more precisely you have to think about whether you want to allow such packings, and if not, what extra conditions you want to impose to exclude them.

Note that in the case |X|=|Y|=|Z|=2, constraining the 2-marginals is the same as constraining the moments of degree <=2, so the log of the maximum entropy distribution is a degree-2 polynomial. Solving in practice is then an exercise in root-finding (solving for the coefficients of the polynomial that give the desired marginals/moments).

Seems to have some similarity to the sofa moving problem, so I would expect methods from that problem to be useful here

think about the case of oblate and prolate spheroids for the two questions

you're welcome. Feel free to write up an answer to #2 if you think it adds value

and @alvarezpaiva's comment on my answer to that question should give you a good hint for question #2, if you replace the square of the radial function by the support function

your question #1 seems to be a duplicate of mathoverflow.net/questions/138525/…

In an answer to a post linked in the question (mathoverflow.net/questions/120240), Günter Rote seems to claim "The best Lipschitz constant (for the Hausdorff distance) is achieved by the so-called Steiner point". Is that a different constant than what is discussed here?

This is not the appropriate venue for this question, but I believe what you're looking for is just a gradient-free optimization method? Many numerical libraries implement one or more such methods (e.g. scipy has a Nelder-Mead routine). Maybe start here: en.wikipedia.org/wiki/Derivative-free_optimization

Your conjecture in the update seems to be a known open problem. See the references and improved upper bound in this answer: mathoverflow.net/questions/173712/…