High-precision linear minimization is no slower than projection

Abstract 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 $$\varepsilon $$ -approximate linear minimization takes at least $$L(\varepsilon )$$ real vector-arithmetic operations and projection requires P operations, then $$\mathcal {O}(P)\ge \mathcal {O}(L(\varepsilon ))$$ is guaranteed. This concept is expounded with examples, an explicit error bound, and an exact linear minimization result for polyhedral sets.