 Uniformly convex variational integrand

A gradient integrand is uniformly convex in this elliptic variational sense when its Hessian matrix in the gradient variables has a common positive lower bound. Then $(F_p(x,p)-F_p(x,q))\cdot (p-q)\ge \lambda |p-q{|}^2$. Averaging $F_{pp}$ along a line segment in gradient space preserves the same lower bound.