Intersect the ambient balls with . All points on the vertical arm have distance at least from the centre, so the open ball is
(1)
The closed ball also contains the endpoints of that segment and the lower half of the vertical arm:
(2)
The closure of the open ball in is just . Both endpoints are limits of points in the open segment, whereas a vertical point of positive height has a small enough neighbourhood disjoint from it. Thus every point with witnesses strict containment.
Figure 1.
Ambient and relative maximum-norm balls
. The square on the left is the ambient closed ball. On the right, the open ball is the horizontal segment without its endpoints; its closure omits the additional vertical part of the relative closed ball. Details.

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