Inscribing Spheres in Topologically Embedded Simplices

In plane geometry, it is well-known that one can inscribe a circle in any triangle. A less trivial yet natural question could be asked about inscribing circles in topologically embedded triangles. This paper presents a proof for the following general theorem: Given any boundary of an n-simplex topologically embedded in Euclidean k-space, with n≤k, there exists a (k−1)-sphere whose interior is disjoint from the embedded boundary of the simplex and which touches each (n−1)-dimensional face on the boundary of the simplex at least once. We say this sphere is inscribed in the embedding. This paper also shows that ellipsoids, cubes, and octahedrons can be inscribed in any topologically embedded boundary of the n-simplex. The uniqueness of inscribed spheres is also discussed.