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In the $d$-dimensional cow-path problem, a cow living in $\mathbb{R}^d$ mustlocate a $(d - 1)$-dimensional hyperplane $H$ whose location is unknown. Theonly way that the cow can find $H$ is to roam $\mathbb{R}^d$ until itintersects $\mathcal{H}$. If the cow travels a total distance $s$ to locate ahyperplane $H$ whose distance from the origin was $r \ge 1$, then the cow issaid to achieve competitive ratio $s / r$.  It is a classic result that, in $\mathbb{R}^2$, the optimal (deterministic)competitive ratio is $9$. In $\mathbb{R}^3$, the optimal competitive ratio isknown to be at most $\approx 13.811$. But in higher dimensions, the asymptoticrelationship between $d$ and the optimal competitive ratio remains an openquestion. The best upper and lower bounds, due to Antoniadis et al., are$O(d^{3/2})$ and $\Omega(d)$, leaving a gap of roughly $\sqrt{d}$. In thisnote, we achieve a stronger lower bound of $\tilde{\Omega}(d^{3/2})$.

A Nearly Tight Lower Bound for the $d$-Dimensional Cow-Path Problem