Finite-dimensional linear minimization is no slower than projection

Let $\mathcal H$ be the Hilbert space underlying the compact convex set considered, and let $P$ denote the optimal runtime complexity of its projection operator, while $L(0)$ denotes the optimal runtime complexity of exact linear minimization. Finite-dimensional complexity claim. If the dimension of $\mathcal H$ is finite, then

$$L(0)=\mathcal{O}(P).$$

This establishes that exact linear minimization is no slower than projection up to a constant factor in finite-dimensional spaces. The surrounding discussion notes that the general relationship between $P$ and $L(0)$ warrants further investigation, including whether some compact convex set could satisfy $P<L(0)$; the supplied text does not state whether this finite-dimensional claim has been proved.