minima function is positive defined

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Is a function: C(x, y) = min(x, y) a positive definite function?

I think it is not positive - a defined function. If values x and y are negative the f (x, y) <0

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Well, if you mean $C(x,x)\geq 0$ with equality iff $x=0$, then certainly it isn't positive definite. Pick $x=-1$ then you habe $C(x,x)=-1<0$ a contradiction.