High-precision linear minimization is no slower than projection

This note demonstrates that, for all compact convex sets, high-precision linear minimization can be performed via a single evaluation of the projection and a scalar-vector multiplication. In consequence, if ε -approximate linear minimization takes at least L(ε ) real vector-arithmetic operations and projection requires P operations, then O(P)≥ O(L(ε )) is guaranteed. This concept is expounded with examples, an explicit error bound, and an exact linear minimization result for polyhedral sets.