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LetX(-ϱB  m C  n  be a compact set over the unit sphere ϱB  m  such that for eachz∈ϱB  m  the fiberX  z ={ω∈C  n ;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain inC  n . The polynomial hull $$\hat X$$ ofX is described in terms of the Perron-Bremermann function for the homogeneous defining function ofX. Moreover, for each point (z 0,w 0)∈Int $$\hat X$$ there exists a smooth up to the boundary analytic discF:Δ→B  m C  n  with the boundary inX such thatF(0)=(z 0,w 0).

Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration