The higher-dimensional Erdős distinct-distances conjecture

For every fixed integer d ≥ 3, we prove that every set of n ≥ 2 distinct points in ℝ${}^d$ determines at least $c_d{n}^{2/d}$ distinct distances, where $c_d>0$ depends only on d. This resolves the higher-dimensional Erdős distinct-distances conjecture positively.