Quasiconvexity of the Burkholder function on symmetric matrices

What the math gives to ML

The paper gives an explicit nonconvex integrand on 2x2 symmetric matrices whose integral over a Hessian perturbation is nevertheless nonnegative: this is a constructive quasiconvexity certificate rather than a generic convex penalty. The transferable object is a curvature regularizer for scalar fields on 2D grids, where the Hessian is automatically symmetric and the theorem rules out energy-lowering oscillatory perturbations under fixed boundary values. A practical first use is to replace or augment squared-Hessian penalties in image networks with this Burkholder energy, while testing whether it preserves sharp anisotropic structure and improves robustness without producing checkerboard artifacts.