[dchftcs]

Right, indeed you need to first preserve the origin, but also that is trivially true for a linear map like JL.

[srean]

As far as I can recall JL holds for affine transformations too, in any case it's an existence result. Have to double check on the affine bit.

The popular proof does uses random linear transforms and they indeed will not change the origin, but that's just one class of transforms with the JL property.