A 2D Voronoi-diagram data structure answers smallest-enclosing-disk rectangle queries in O(log^4 n) deterministic time after O(n log^2 n) preprocessing.
6 Franz Aurenhammer, Rolf Klein, and Der-tsai Lee.Voronoi Diagrams And Delaunay Triangu- lations
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Smallest Enclosing Disk Queries Using Farthest-Point Voronoi Diagrams
A 2D Voronoi-diagram data structure answers smallest-enclosing-disk rectangle queries in O(log^4 n) deterministic time after O(n log^2 n) preprocessing.