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Suppose I have three coordinates functions of a vector of parameters: x[q],y[q],z[q] where q is at least a three dimensional vector in some domain Q. Is it possible to plot in 3D the space spanned by q? By that I mean all the triplets {x',y',z'} such that there is at least one q in Q such that x[q]=x',y[q]=y',z[q]=z'? Of course I would be happy if this worked with intervals (say Q is the unit cube). And of course I would be happy with the surface of the subset.

I need to quickly check that this is a convex subset of R^3. So any other method is welcome.

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Assuming your three scalar functions x, y, and z have been defined, you'd do something like this:

n = 3;
q = Array[q1, n]; 
RegionPlot3D[
 Length[FindInstance[(x @@ q == x1) && (y @@ q == y1) && (z @@ q == z1), q]] > 
  0, {x1, -1, 1}, {y1, -1, 1}, {z1, -1, 1}]

Here, n is the dimension of $Q$. This is all I can do without knowing how the unknowns in your question are actually defined.

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