In this paper, we develop frequency-domain phase noise estimation schemes for an orthogonal frequency division multiplexing (OFDM) system that utilize the geometrical structure associated with the spectral components of the complex exponential of the phase noise process. This geometry can be expressed as a set of non-convex quadratic equality constraints that involve permutation matrices. Since the oscillator power spectral density is typically a low-pass process, we can reduce the dimensionality of the phase noise spectral vector. In particular, we propose a new dimensionality reduction model that preserves the phase noise spectral geometry when moving from smaller to larger dimensional spaces. Using this new model and building upon existing works, the phase noise estimation problem is posed as an optimization problem. We show, using the S-procedure, that the optimization problem has an optimal solution that is attained and can be solved using semidefinite programming. In order to reduce the computational complexity of the aforementioned optimization problem, we consider sub-optimal schemes that provide an estimate satisfying this geometry. Through numerical simulations, we demonstrate superior bit-error-rate performance when the proposed model used in conjunction with the phase noise spectral geometry is incorporated in the estimation process.