Approximation of convex figures by pairs of rectangles

Abstract

We consider the problem of approximating a convex figure in the plane by a pair (τ, R) of homothetic (i.e. similar and parallel) rectangles with τ⊂C⊂R. We show the existence of such pairs where the sides of the outer rectangle have length at most double the length of the inner rectangle, thereby solving a problem posed by Pólya and Szegő.

If the n vertices of a convex polygon C are given as a sorted array, such an approximating pair of rectangles can be computed in time O(log³ n).

This research was supported by the Deutsche Forschungsgemeinschaft under Grant Al 253/1-1, Schwerpunktprogramm “Datenstrukturen und effiziente Algorithmen”