Lower-dimensional limits of cubic Lovelock gravity

We obtain the lower-dimensional limits $\left(p=2,3,4,5,6\right)$ of cubic Lovelock gravity through a regularized Kaluza-Klein reduction. By taking a flat internal space for simplicity, we also study the static black hole solutions in the resulting theories. We show that the solutions match with the ones obtained from the "naive limit" of D-dimensional equation for the metric function, which is obtained by first scaling the relevant couplings by a factor of $\frac{1}{D-p}$ and then taking the limit $D\to p$, with one important exception: In 4D, one obtains the expected solution only for the black hole with a planar horizon.