Division of $n$-Dimensional Euclidean Space into Circumscribed $n$-Cuboids

Vladimir Dragović ; Roger Fidèle Ranomenjanahary

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected issues of mathematics and mechanics, Tome 310 (2020), pp. 149-160

Résumé

In 1970, Böhm formulated a three-dimensional version of his two-dimensional theorem that a division of a plane by lines into circumscribed quadrilaterals necessarily consists of tangent lines to a given conic. Böhm did not provide a proof of his three-dimensional statement. The aim of this paper is to give a proof of Böhm's statement in three dimensions that a division of three-dimensional Euclidean space by planes into circumscribed cuboids consists of three families of planes such that all planes in the same family intersect along a line, and the three lines are coplanar. Our proof is based on the properties of centers of similitude. We also generalize Böhm's statement to the four-dimensional and then $n$-dimensional case and prove these generalizations.