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.)
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You can let $$$D\to D-C$$$
C is variable across functions...
Oh, sorry, I misunderstood you — you can just binary search on the min and then solve a linear program.
for that problem, I have to do a lot of queries, so $$$O(qn)$$$ wouldn't cut it. Think of the Euclidean closest pair problem.
If the function set is fixed, just precompute the minimum.
If a new function is inserted each time, this is a dynamic 2D convex hull — split the hull into an upper hull and a lower hull, and maintain each with a balanced tree or
std::set.