Searching for the closest-pair in a query translate

We consider a range-search variant of the closest-pair problem. Let Γ be a fixed shape in the plane. We are interested in storing a given set of n points in the plane in some data structure such that for any specified translate of Γ, the closest pair of points contained in the translate can be reported efficiently. We present results on this problem for two important settings: when Γ is a polygon (possibly with holes) and when Γ is a general convex body whose boundary is smooth. When Γ is a polygon, we present a data structure using O(n) space and O(log n) query time, which is asymptotically optimal. When Γ is a general convex body with a smooth boundary, we give a near-optimal data structure using O(n log n) space and O(log² n) query time. Our results settle some open questions posed by Xue et al. [SoCG 2018].