Mean-curvature comparison at tangential contact
= Mean-curvature comparison at tangential contact
Suppose two graph surfaces $z=f(x,y)$ and $z=g(x,y)$ are tangent at the origin, are oriented upward, and $g\geq f$ nearby. Then $g-f$ has a local minimum, so its <Hessian matrix> is <positive semidefinite matrix>[positive semidefinite]. At the common horizontal tangent plane,
$$
H_g-H_f=\frac12\operatorname{tr}\operatorname{Hess}(g-f)\geq0.
$$