The geometry of stable discrete polynomials using their coefficients and reflection coefficients is investigated. Two linear Schur invariant transformations with a free parameter in the polynomial coefficient space are introduced. The first transformation R-n X R -> R-n maps an arbitrary stable polytope into another stable polytope. The second transformation R-n X R -> Rn+1 maps a stable tilted n-dimensional hyperrectangle defined by the discrete Kharitonov theorem into a stable (n + 1)-dimensional polytope.