Estimating axial symmetry using random projections

This paper studies the problem of identifying directions of axial symmetry in multivariate distributions. Theoretical results are derived on how the measure or cardinality of the set of symmetry directions relates to spherical symmetry. The problem is framed using random projections, leading to a proof that in ${\text{RR}}^2$, agreement on two random projections is enough to identify the true axes of symmetry. A corresponding result for higher dimensions is conjectured. An estimator for the symmetry directions is proposed and proved to be consistent in the plane.