Tightly Embedded 2-dimensional Polyhedral Manifolds
American Journal Mathematical, 1965, pp. 462-472.
This paper introduces the word "tight" to describe the geometric
condition that corresponds in the smooth case to minimal total
absolute curvature. For any two-dimensional surface, an embedding is
tight if almost any height function has exaclty one maximum when
restricted to the surface. This is equivalent to the fact that every
local support hyperplane is a global support hyperplane. Although any
tightly embedded smooth surface must be contained in some
five-dimensional subspace, there is not such restriction for
polyhedral surfaces:
Theorem A: For any n, there is a two-dimensional polyhedral surface
M(n) embedded in R^n and not lying in any affine
(n-1)-dimensional subspace.
In order to find examples for laarge n, it is necessary to use
surfaces with high genus.
If M is a polyhedral surface tightly embedded in R^n, then n <
(7 - 24 Euler characteristic of (M))/2.