[hansvm]

I thought about it a bit more:

- That last point (attaining sqrt(2)) follows from Bolzano-Weierstrass. Just unroll the embedding into an n^2-dimensional space if you have n points, and that sequence of embeddings (as a technical detail, center them on the origin to squeeze them all into a bounded space). The function (skew of that embedding) of the limit equals the limit of the function in this case, so there is some embedding attaining that bound.

I'm still not exactly sure how the result I'm basing this all on is proven.