Extensions of bounded holomorphic functions on the tridisk

A set V in the tridisk D³ has the polynomial extension property if for every polynomial p there is a function φ on D³ so that ||φ||_D3 = ||p||_V and φ|_V = p|_V. We study sets V that are relatively polynomially convex and have the polynomial extension property. If V is one-dimensional, and is either algebraic, or has polynomially convex projections, we show that it is a retract. If V is two-dimensional, we show that either it is a retract, or, for any choice of the coordinate functions, it is the graph of a function of two variables.