Achieve O(k n^{1+ε}) time for largest-similar-copy containment

Determine whether the problem of computing the largest similar copy of a convex k-gon P contained in a convex n-gon Q in the plane under translation, rotation, and scaling can be solved in O(k n^{1+ε}) time for every fixed constant ε > 0, thus strictly improving the previous O(k n^2) bound for all k.

Background

The paper revisits the classic problem of placing the largest similar copy of a convex polygon P inside another convex polygon Q. Earlier algorithms required at least quadratic time in n. The authors present a near-linear algorithm for triangles and extend it to general k with running time O(k^{O(1/ε)} n^{1+ε}) for any ε > 0.

They explicitly leave open whether the general running time can be improved to O(k n^{1+ε}) for all constant ε > 0, which would remove the superpolynomial-in-1/ε dependence on k and provide a strict improvement over the longstanding O(k n^2) bound for all k.