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 [More]. When equality holds, the mapping is called tight.

This definition turns out to have several important geometric consequences and equivalent definitions:

[Right] Tightness and the convex hull.
[Right] Tightness and polar height functions.
[Right] Tightness and the two-piece property.
[Right] Tightness and homology: the modern definition.
[Right] Tightness for polyhedral surfaces.

[Left] Kuiper's original question
[Up] Introduction

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7/21/94 dpvc@geom.umn.edu -- The Geometry Center