You may want to try stats.stackexchange.com

Note that "undecidable" has two meanings that should not be confused: "independent of axioms" and "not computable".

A good lemma could be that if a sample leads to $\mu$ not being in the convex hull, translate the sample (maybe add $\mu-c$ where $c$ is the centroid of the sample) and argue that the joint pdf $f_{X_1,\ldots,X_n}$ thereby increases.

Oh and the distance to the origin of the plane $x+y+z=2$ is $2/\sqrt{3}$, whereas $v$ has length $1<2/\sqrt{3}$, so this also answers the second question.

After 5 hours we both answer within the same minute! :)

"have non-empty intersection with every subset of $W$" -- did you mean "every element of $W$"?

So meta - meta - mathematical in intent is a little bit like physical in intent or financial in intent... that makes sense.

Is that a typo, "$P(X_{t-})=0$"?

I read somewhere that meta.meta.mathoverflow questions should go to meta.stackoverflow.com, so again we have collapse to the first level.

That's good, how about a decidable example...

Hi, regarding "every element of $\mathcal{M}$ is definable but there is no single constant bounding the quantifier depth of the defining formulae" -- do you have an example of such an $\mathcal M$"?

The construction of the problem seems to rely heavily on the "coincidence" $2^4=4^2$... :)