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, Convex figure, Rectangle, Schuettelung, Separation of convex sets, Theorem of moreau-rockafellar
dx.doi.org/10.1007/s11590-015-0941-0, hdl.handle.net/1765/87969
Optimization Letters
Department of Econometrics

Brinkhuis, J. (2016). Inner and outer approximation of convex sets using alignment. Optimization Letters, 10(7), 1403–1416. doi:10.1007/s11590-015-0941-0