Local Nonconvexity Indices for $C^{1,1}$ Functions via Generalized Hessians

Davydov, Moldavskaya, and Zitikis introduced local indices for quantifying the lack of convexity of a $C^2$ function by measuring the nuclear-norm distance of its Hessian from the cone of positive semidefinite matrices. This paper develops a local analogue for functions of class $C^{1,1}$. At a point $x$, the classical Hessian is replaced by the Clarke-type generalized Hessian set $\text{Hess}(h;x)$, defined as the closed convex hull of limiting Hessians at nearby twice differentiability points. Evaluating the same spectral functional over $\text{Hess}(h;x)$ gives an interval-valued local nonconvexity index whose lower and upper endpoints represent, respectively, the least and greatest visible second-order nonconvexity at $x$. The construction reduces to the original smooth index when $h\in C^2$, vanishes for convex $C^{1,1}$ functions, is invariant under orthogonal changes of variables, satisfies a subadditivity inequality for the upper endpoint under sums, and is upper semicontinuous in its upper endpoint. We also relate the upper endpoint to a pointwise weak-convexity curvature modulus and give explicit $C^{1,1}\setminus C^2$ examples. The paper is deliberately local in scope: it proposes a scalar diagnostic extracted from generalized Hessian sets, not a replacement for the richer second-order variational theory of nonsmooth convexity.