Inner and outer approximation of convex sets using alignment |
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Authors: | Email author" target="_blank">Jan?BrinkhuisEmail author |
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Institution: | 1.Econometric Institute, Erasmus School of Economics,Erasmus University Rotterdam,Rotterdam,The Netherlands |
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Abstract: | 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. |
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