Mixed linear regression is a high dimensional affine space clustering problem where the goal is to find the parameters of multiple affine spaces that best fit a collection of points. We introduce a convex 2nd order cone program (based on l1/fused lasso) which allows us to reformulate the mixed linear regression as an Rd clustering problem. The convex program is parameter free and does not require prior knowledge of the number of clusters, which is more tractable while clustering in Rd. In the noiseless case, we prove that the convex program recovers the regression coefficients exactly under narrow technical conditions of well-separation and balance. We demonstrate numerical performance on BikeShare data and music tone perception data.