The binary reflected Gray code orders the vertices of the unit hypercube along a Hamiltonian path in which consecutive vertices differ in exactly one coordinate. While Gray codes have been extensively studied from a combinatorial perspective, much less is known about the polyhedral structure of convex hulls of contiguous subpaths of this order. This paper develops an exact linear description for Gray intervals based on recursive projections on lower-dimensional subintervals and simple lifting operators. We also study the separation problem for the recursive description, which can be performed in polynomial time by a dynamic program over reachable subintervals. Finally, we specialize the recursion to prefix and suffix Gray intervals and derive compact closed-form descriptions in terms of the binary expansions of the endpoints.