Abstract
Communicated by R. H. Bing, May 20, 1959 Let E be a euclidean ^-space with a rectangular cartesian coordinate system (x) â (xi, ⢠⢠⢠, xn), and let (y) be any system which is a rotation of (x). Let AQE be a closed bounded set containing n + 1 linearly independent points. Its circumscribed (y)-box is the set diuyi^bi ( i= 1, ⢠⢠⢠, n) where ai and bi are the respective minimum and maximum values of ^ on 4 . Let d = bi â ai be interpreted as a function on the space Rn-i of rotations of coordinate systems, which is also the rotation space of the unit (w--l)-sphere S~C_E. L e t / : Rn-iâ>E be the function which maps r G ^ n i onto the point (ci(r), ⢠⢠⢠, cn(r)), relative to the fixed initial coordinate system (x). Let D be the diagonal Xi â ⢠⢠⢠=xn in E . The circumscribed (y) -box corresponding to a point rÂŁi? f t_i is an n-cube if and only if / ( > ) ÂŁ P . Accordingly, K=f~(D), a subspace of Rn-i, will be called the space of circumscribed n-cubes of A. Its structure can be studied by means of the mapping Ć. For the purpose of this study the significant properties are as follows: (1) Ć is a continuous mapping of Rn-i into the region X;>0 (i= 1, ⢠⢠⢠, n) of E (2) f(Rn-i) is symmetric with respect to D. This second property follows from the fact that all possible permutations of axial directions can be achieved in a symmetric way through rotations. There is no need to distinguish between the two possible senses on a given ^-direction, since the value of Ci is the same for both. Hence, one gets odd as well as even permutations of the c's. Let T~ be the simplex in E with vertices at the unit points on the (x)-axes. A central projection from the origin carries the mapping Ć into a continuous mapping g: Rn-iâ^T ~~ where g(Rn-i) is symmetric in the barycentric coordinates on T~. The inverse image g^)Âť where q is the barycenter of P 1 1 , is identical with f~(D) =K. This leads to the following result.
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