So we have a counterexample to the triangle inequality. Basically, the main reason these are not norms is that the unit ball is not convex, which means you can pick two points in the unit ball, like (0,1) and (1,0), and draw a line between them and have points on that line with norm greater than one: for example ||(1/2, 1/2)|| =(2/2p)1/p = 2(1
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