I'm sure that someone out there in computational-geometry-land has solved this problem, but I was unable to find an answer so I thought I'd post my question here:
Given two positive quadratic functions a1x^2 + b1x + c1 and a2x^2 + b2x + c2, find a third quadratic function which is a "greatest" lower-bound on both these two functions.
I'm deliberately leaving the term "greatest" vague, but intuitively, I want the third quadratic to hug the other two functions as closely as possible (but of course not exceed them for any x). Also, x is a vector. One definition of "greatest" might mean that the area between the third function and the other two is minimized. There might be other definition that make more sense.
I'm hoping there's an analytical solution to this problem.