[ska]

You may find it interesting to read about Lp norms, and their relationship to inner products on vector spaces. I think the OP is mixing up norm and inner product terminology. This happens often because you derive an norm from any inner product, but the other way may not exist.

If you plot on the plane the distance = 1 line, then L_1 gives you a diamond, L_2 a circle, L_inf a square. [More precisely, the unit circle under the related metric (distance function) looks like those euclidean shapes]