Tightness and Height Functions

For z\in S^2 a unit vector in \R^3, the function zf\colon M\to\R given by

zf(p) = z\cdot f(p)

is the height function on M in the direction of z.

Theorem: f is tight if, and only if, every non-degenerate height function on M has exactly one maximum and one minimum.

Consequence: All the positive curvature of M is on the surface of the convex hull of f(M), denoted \dHf.