Burning is a discrete-time model for propagation in which a new fire starts in each round, while each existing fire expands by one unit of distance along the underlying metric. In geometric burning, the input is a finite point set, and the goal is to burn all points in as few rounds as possible. Equivalently, burning a point set in k rounds corresponds to covering it with metric balls of distinct radii in {0,1,…,k−1}; the objective is to minimize k. Previous work has studied the problem mainly under the Euclidean metric. In this paper, we study geometric burning under the
Nearby in the stack
L1
and
L∞
metrics. The problem remains NP-hard in both settings. The
L1
and
L∞
metrics provide additional geometric structure, which allows us to obtain improved approximation guarantees, especially for anywhere burning. We first present a simple
(2+ε)
-approximation for both anywhere burning and point burning. We then improve the anywhere burning approximation to
7/4+ε=1.75+ε
, and give a
(3151/1620+ε)
-approximation for point burning, where
3151/1620<1.9451
. We also extend the anywhere burning result under
L∞
to every fixed dimension
d≥3
to achieve a
(2−2d+11+ε)
-approximation. Finally, using standard comparisons between planar
Lp
distances, we transfer our
L1
and
L∞
algorithms, together with known Euclidean burning algorithms, to obtain approximation guarantees for every fixed