Inner and outer approximation of convex sets using alignment

We show that there exists, for each closed bounded convex set C in the Euclidean plane with nonempty interior, a quadrangle Q having the following two properties. Its sides support C at the vertices of a rectangle r and at least three of the vertices of Q lie on the boundary of a rectangle R that is a dilation of r with ratio 2. We will prove that this implies that quadrangle Q is contained in rectangle R and that, consequently, the inner approximation r of C has an area of at least half the area of the outer approximation Q of C. The proof makes use of alignment or Schüttelung, an operation on convex sets

Approximation of convex figures by pairs of rectangles

We consider the problem of approximating a convex figure in the plane by a pair {r; R} of homothetic �that is, similar and parallel � rectangles with r � C � R. We show the existence of such a pair where the sides of the outer rectangle are at most twice as long as the sides of the inner rectangle, thereby solving a problem posed by Polya and Szegö. 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(log² n)

Approximation of Convex Figures by Pairs of Rectangles

We consider the problem of approximating a convex figure in the plane by a pair #r;R# of homothetic #that is, similar and parallel# rectangles with r # C # R.We show the existence of such a pair where the sides of the outer rectangle are at most twice as long as the sides of the inner rectangle, thereby solving a problem posed byP#olya and Szeg#o. 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(log²n)