a norm is a total size or length of all vectors in a vector space or matrices. - Rorasa’s blog / wikipedia

Closest point in L1 vs L2

In L1 (Manhattan):(1,1) and (2,0) have the same distance to (0,0) and cannot be distinguished.
In L2 (1,1) would be the one and only closest point.

see also

• Differences between the L1-norm and the L2-norm (Least Absolute Deviations and Least Squares)