In my failed attempts at trying to reinvent the Euclidean Closest Pair Problem, I have mostly gotten in the road of trying to check for a list of functions $$$f(x,y)=A\times x+B\times y+C$$$ whether the minimum is larger than a value $$$D$$$ or not.

Can this problem be solved quickly? (For the case $$$C=0$$$, this is just CHT.)