>>11198170>>11200987OP's pic has ||x|| which is used to label norms but if you want a topological approach we can consider open (or closed) balls in R^2.
An open ball of radius r is the set of points less than a distance r from some point x in a metric space M
B_r (a) = {b : d(a,b) < r}
A closed ball is every point within a distance r including points that are distance r from the "center"
We define the distance d(a,b) using a metric, which a function that takes two points on a set (in this case R^2) and returns a positive distance d.
A metric must also meet the following requirements (you can verify each metric meets these yourself)
1. If a=b then d(a,b)=0. A distance from a point to itself is 0
2. d(a,b) = d(b,a). A metric is symmetric
3. d(a,c) =< d(a,b) + d(b,c) the triangle inequality
Unsurprisingly the usual metric d((a,b),(c,d)) = sqrt((c-a)^2+(d-b)^2) yields a ball that looks like a circle and in some sense we can consider balls under different metrics to be a circle under that metric.
The taxi cab metric d((a,b),(c,d)) = |c-a| + |d-b| gives the diamond
And d((a,b),(c,d)) = max(|c-a|,|d-b|) yields a square
Another interesting metric to consider is d((a,b),(c,d))=1 if (a,b)=(c,d) and d((a,b),(c,d))=0 if (a,b)=/=(c,d)
