Question

if ||x + y|| = ||x|| + ||y||, then x,y are parallel vectors--true or false? Clearly ||x + v|| = ||x|| + ||v|| if ||*|| is an inner product norm by bilinearity of an inner product space, but I can't think of a counter-example using a non-convex norm like Lp for p = 1/2

Answer 1

In case anybody Googles this in 10 years, the equality doesn't hold for the 1-norm or the infinity-norm because their unit spheres aren't strictly convex. lol whoops