This paper characterizes the geometrical structure of receding horizon control (RHC) of linear, discrete-time systems, subject to a quadratic performance index and linear constraints. The geometrical insights are exploited to derive a closed-form solution for the case where the total number of constraints is less than or equal to the number of degrees of freedom, represented by the number of control moves. The solution is shown to be a partition of the state space into regions for which an analytical expression is given for the corresponding control law. Both the regions and the control law are characterized in terms of the parameters of the open-loop optimal control problem that underlies RHC and can be computed off-line. The solution for the case where the total number of constraints is greater than the number of degrees of freedom is addressed via an algorithm that iteratively uses the off-line solution and avoids on-line optimization.