Maximin location of convex objects in a polygon and related dynamic Voronoi diagrams

This paper considers the maximin placement of a convex polygon P inside a polygon Q, and introduces several new static and dynamic Voronoi diagrams to solve the problem. It is shown that P can be placed inside Q, using translation and rotation, so that the minimum Euclidean distance between any point on P and any point on Q is maximized in O(m⁴nλ16(mn) log mn) time, where m and n are the numbers of edges of P and Q, respectively, and λ16(N) is the maximum length of Davenport–Schinzel sequences on N alphabets of order 16. If only translation is allowed, the problem can be solved in O(mn log mn) time. The problem of placing multiple translates of P inside Q in a maximin manner is also considered.