I have already asked a similar question. Still, I have been thinking about $l^p$ spaces all summer, and I still don't have a grasp of it. I thought $l^p$ spaces are a generalization of Euclidean space. If that is the case, I don't understand the relationship between an Lp space and this famous graph:
https://en.wikipedia.org/wiki/Lp_space#/media/File:Vector-p-Norms_qtl1.svg
If $l^p$spaces are not a generalization of Euclidean space, could someone please tell me what $l^p$ spaces are and how the linked graph motivates understanding?
To all those who may contribute, I thank you, in advance, VERY much for your insight.
The $l^p$ space is a special case of Minkowski space, all these "non-Euclidean "geometries have been derived based on using the first four Euclidean postulates together with various negations of the fifth. take a look at https://www.cambridge.org/core/books/minkowski-geometry/BEB8FE99553CABD2BECD623887C879B8
imagine you are in a space that is not homogenous, so distance is not the same in every direction. try to have a look on the spherical geometries and also the hyperbolic plane.
in the Euclidean plane the minimal distance between two points is just the segment when you are in $l^p$ space the minimal distance are arcs of the $l^q$ disk passing through the two points (such that l^q is the dual of l^p)
in general, when your unit disk is the euclidean disc everything is Clair (humain are euclidean) you can compute angles, area, perimeter, curvature the orthogonality between two vectors ... when your unit disk is just a convex body you lose a lot of things, first the symmetry of the orthogonality ....