We consider the oracle complexity of constrained convex optimization given access to a Linear Minimization Oracle (LMO) for the constraint set and a gradient oracle for the $L$-smooth, $L$-strongly convex objective. This model includes Frank-Wolfe methods and their many variants. Over the problem class of $\alpha$-strongly convex constraint sets $S$, we demonstrate that one can construct hard “zero-chain” instances in the classical style of Nemirovski and Yudin. From our new approach to adversarial oracle construction, we prove that no such deterministic method can guarantee a final objective gap less than $\varepsilon$ in fewer than $\Omega(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$ iterations. Our lower bound partly matches the accelerated Frank-Wolfe theory of Garber and Hazan (2015) of $O(\sqrt{L(\mathrm{diam}(S)^2+1/\alpha^2)/\varepsilon})$. Second, we consider optimization over $\beta$-smooth sets, finding that in the modestly smooth regime of $\beta=\Omega(1/\sqrt{\varepsilon})$, no complexity improvement for span-based LMO methods is possible against either compact convex sets or strongly convex sets.