POLYGONS CIRCUMSCRIBED ABOUT CLOSED CONVEX CURVES

Both proofs allow extension to more than four sides. The first proof, by means of the Poincaré ring theorem [l] (*), is restricted to the plane but allows somewhat more general subdivision of the sides than simple bisection. The second proof, by minimal area, extends the theorem to circumscribed polyhedra of any number of sides in any number of dimensions. The need for Theorem 1 first appeared in the course of some proofs of properties of unit spheres in normed linear spaces [2]. The applications to the problem which originally suggested Theorem 1 are clearer if, in a normed linear space, x normal to y is defined to mean ||a;e+y|| ^\\y\\ for all real a; that is, that the line through y parallel to Ox comes no closer to the origin than y. This relationship in a Euclidean space is equivalent to orthogonality of x and y; in other normed spaces it retains some of the properties of orthogonality but is, in general, not symmetric [2, §6]. Theorem 1 asserts that in any twodimensional normed space there exist two points of norm one each normal to the other. The generalization, Theorem 4.1, to « dimensions asserts that in each n-dimensional normed linear space there exist n points x¡ such that ¿2i^ia¡Xj is normal to x,for any choice of i = \, ■ ■ ■ , n and real numbers a/. Translating this notion of normality in the obvious way to Finsler spaces gives : Through every point P of an n-dimensional locally-Minkowskian Finsler space there exist w hypersurfaces each normal at P to the intersection of all the others. 2. Proof by the ring theorem. Throughout this section assume that C is a closed convex (not necessarily symmetric) curve with no flat sides; that is, that no line segment of positive length lies on C. (The general case of Theorem 1 is easily proved by approximation from this special case.) Choose a positive direction, say counterclockwise, around C. Then for each point b outside or on C there is a unique tangent line from b to C which passes in the chosen