##
High Codimensional 0-Tight Maps on Spheres

by Thomas F. Banchoff

The condition of 0-tightness for smooth immersions of 2-manifolds in E^M
is equivalent to minimal total absolute curvature, but these notions
do not coincide in higher dimensions. By a result of Chern and
Lashof, if a smooth n-sphere embedded in E^M has minimal total
absolute curvature, then it must bound a convex (n+1)-cell in an affine
(n+1)-dimensional subspace. We will show that for any n > 2 and
M > n, there exists a 0-tight polyhedral embedding of the n-sphere into
E^M whose image lies in no hyperplane.