The problem of robust mean estimation in high dimensions is studied, in which a certain fraction (less than half) of the datapoints can be arbitrarily corrupted. Motivated by compressive sensing, the robust mean estimation problem is formulated as the minimization of the ℓ0-'norm' of an outlier indicator vector, under a second moment constraint on the datapoints. The ℓ0-'norm' is then relaxed to the ℓp-norm (0 < p ≤ 1) in the objective, and it is shown that the global minima for each of these objectives are order-optimal and have optimal breakdown point for the robust mean estimation problem. Furthermore, a computationally tractable iterative ℓp-minimization and hard thresholding algorithm is proposed that outputs an order-optimal robust estimate of the population mean. The proposed algorithm (with breakdown point ≈0.3) does not require prior knowledge of the fraction of outliers, in contrast with most existing algorithms, and for p = 1 it has near-linear time complexity. Both synthetic and real data experiments demonstrate that the proposed algorithm outperforms state-of-the-art robust mean estimation methods.