  In the $d$-dimensional cow-path problem, a cow living in $\mathbb{R}^d$ must
locate a $(d - 1)$-dimensional hyperplane $H$ whose location is unknown. The
only way that the cow can find $H$ is to roam $\mathbb{R}^d$ until it
intersects $\mathcal{H}$. If the cow travels a total distance $s$ to locate a
hyperplane $H$ whose distance from the origin was $r \ge 1$, then the cow is
said 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 is
known to be at most $\approx 13.811$. But in higher dimensions, the asymptotic
relationship between $d$ and the optimal competitive ratio remains an open
question. 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 this
note, we achieve a stronger lower bound of $\tilde{\Omega}(d^{3/2})$.
