I'm trying to work through the math here, and I don't understand why these two propositions are equivalent:
1)
min_{x,y} |x-y|^2
x ∈ A
y ∈ B
2)
= min_{x,y} d
d ≥ |x-y|^2
x ∈ A
y ∈ B
What is 'd'? If d is much greater than |x-y|^2 at the actual (x, y) with minimal distance, and equal to |x-y|^2 at some other (x', y'), couldn't (2) yield a different, wrong solution? Is it implied that 'd' is a measure or something, such that it's somehow constrained or bounded to prevent this?
1) min_{x,y} |x-y|^2
2) = min_{x,y} d What is 'd'? If d is much greater than |x-y|^2 at the actual (x, y) with minimal distance, and equal to |x-y|^2 at some other (x', y'), couldn't (2) yield a different, wrong solution? Is it implied that 'd' is a measure or something, such that it's somehow constrained or bounded to prevent this?