Because of the paucity of lower bounds, there has been some work connecting one lower bound to another, as-yet unknown one. I am thinking here of the 3SUM problem: Given $n$ integers, do three sum to zero? The only upper bound (in the most general model) is $O(n^2)$. Jeff Erickson proved a matching lower bound in a restricted linear decision tree model: "Bounds for Linear Satisfiability Problem." Many problems in computational geometry can be reduced to 3SUM, and so are 3SUM-hard. For example: Given a set of $n$ points in the plane (with integer coordinates), find the minimum area triangle with corners in the set.