Andrey Savov wrote: >Ray Vickson wrote: >> >>I would ask: why do you want to minimize the sum of squares? >>For Euclidean distance, that F(x) has some physical and >>statistical meaning, and furthermore leads to a simple >>solution. However, for other norms such as d(x,y) = |x|+|y| >>or d(x,y) = max(|x|,|y|), or for a p-norm with 1 < p < 2, >>what significance can one attach to the sum of squares? >>Certainly it makes _some_ problems much harder instead of >>easier (for example, when d(x,y) = |x| + |y|). > >The norms I had in mind were actually much nicer than the >ones you mention.
Actually, _you_ were the one who mentioned the taxicab norm when I asked for a concrete example of a norm other than the standard Euclidean one to be used as a test case to discuss the questions you raised.
>They were continuous and even convex functions on a subset >of R^n, so for them that point has meaning similar to the >Euclidean norm.
Let's see an example.
>I over-generalized when I posted the question.
So why not try to fix it?
Clarify your assumptions and state what you think is true based on those assumptions.