Circumscribed cubes in euclidean 𝑛-space

Abstract

Communicated by R. H. Bing, May 20, 1959 Let E be a euclidean ^-space with a rectangular cartesian coordinate system (x) — (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 — 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-i—>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 — • • • =xn in E . The circumscribed (y) -box corresponding to a point r£i? f t_i is an n-cube if and only if / ( > ) £ 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 ƒ. For the purpose of this study the significant properties are as follows: (1) ƒ 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 ƒ into a continuous mapping g: Rn-i—^T ~~ where g(Rn-i) is symmetric in the barycentric coordinates on T~. The inverse image g^)» where q is the barycenter of P 1 1 , is identical with f~(D) =K. This leads to the following result.