Tightness and its consequences:
Classically, tightness is defined in terms of the total absolute
curvature integral: given a mapping f of a surface
M into space, the total absolute curvature is
where K is the Gaussian curvature. It can be shown that the
total absolute curvature is always at least 4  X(M) where
X(M) represents the
Euler characteristic of
M
.
When equality holds, the mapping is called tight.
This definition turns out to have several important geometric
consequences and equivalent definitions:

Tightness and the convex hull.

Tightness and polar height functions.

Tightness and the twopiece property.

Tightness and homology: the modern definition.

Tightness for polyhedral surfaces.
Kuiper's original question
Introduction
7/21/94 dpvc@geom.umn.edu 
The Geometry Center