An Inductive Proof for Extremal Simplexes

Abstract

Among the triangles that circumscribe a given circle, the one with minimum area adopts a particularly balanced position: It touches the circle at the midpoints of its sides. This is not surprising, and one may think that it is true due to the symmetry in the circle. It is not! For example, take the circle with its minimal circumscribing triangle and look at their shadow on the wall. Circle turns into an ellipse, the sides of the triangle change size, but they still touch the ellipse at their midpoints. In fact the above proposition remains valid if the circle is replaced by any convex figure. The key element in the proposition is the minimization of the area function, and as such it is reminiscent of extrema problems of calculus. However, if the student routinely rushes to do the problem using the techniques of calculus he or she will soon recognize that the problem is far from being routine. If the convex figure is not smooth, then the usual techniques of calculus that require differentiability of functions will not apply. Using continuity arguments and affine transformations Chakerian and Lange gave an elementary proof of the proposition in this MAGAZINE [2]. Day has proved a generalization of the problem to higher dimensions. He shows that if a polyhedron circumscribes a convex body but does not touch it at the centroid of one of its faces, then there is a polyhedron of smaller volume that also circumscribes the body [3]. The proof involves analysis in space and is not elementary. The reader will find some other, closely related results on this topic in the articles by Bailey [1] and Lange [7, 8]. In this note we will use elementary geometric transformations and cutting and pasting to prove the proposition, and then we will generalize it to n-dimensions by induction. Propositions will be formulated in terms of a convex body, which is a closed and bounded convex set with an interior point in n-dimensional Euclidean space, n > 2.