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 $\dHf$.