Pick's Theorem - What's the Big Deal?

Of course, finding the infimum is computationally harder than answering the question of whether the infimum is less than or equal to some fixed value, but mathematically the problems are equivalent. By a standard compactness argument (what a complex analyst would call a normal family argument), the infimum in (IP) will be attained. By rescaling, we can assume this infimum is 1, and then the Pick matrix (2) cannot be strictly positive definite (for if it were, it would remain so if all the wi s were increased by some factor slightly bigger than one, which would mean by the first part