Compensated Convex Transforms and Geometric Singularity Extraction from Semiconvex Functions

Abstract

We apply upper and lower compensated convex transforms, which are `tight' one-sided approximations of a given function, to the extraction of fine geometric singularities from semiconvex/semiconcave functions and DC-functions in $\mathbb{R}^n$ (difference of convex functions). Well-known examples of (locally) semiconcave functions include the Euclidean distance and squared distance functions. For a locally semiconvex function $f$ with general modulus, we show that `locally' a point is a singular (non-differentiable) point if and only if it is a scale $1$-valley point, and if $x$ is a singular point, then locally the limit of the scaled valley transform exists at every point $x$ and $ \lim_{\lambda\to \infty}\lambda V_\lambda (f)(x)=r_x^2/4$, where $r_x$ is the radius of the minimal bounding sphere of the (Fr\'echet) subdifferential $\partial_- f(x)$ and $V_\lambda (f)(x)$ is the valley transform at $x$. Thus the limit function $\mathcal{V}_\infty(f)(x):=\lim_{\lambda\to+\infty}\lambda V_\lambda (f)(x)=r_x^2/4$ gives a `scale $1$-valley landscape function' of the singular set for a locally semiconvex function $f$, and also provides an asymptotic expansion of the upper transform $C^u_\lambda(f)(x)$ when $\lambda \to \infty$. For a locally semiconvex function $f$ with linear modulus we show that the limit of the gradient of the upper compensated convex transform $\lim_{\lambda\to+\infty}\nabla C^u_\lambda(f)(x)$ exists and equals the centre of the minimal bounding sphere of $\partial_- f(x$, and that for a DC-function $f=g-h$, the scale $1$-edge transform satisfies $\liminf_{\lambda\to+\infty}\lambda E_\lambda (f)(x)\geq (r_{g,x}-r_{h,x})^2/4$, where $r_{g,x}$ and $r_{h,x}$ are the radii of the minimal bounding spheres of the subdifferentials $\partial_- g$ and $\partial_- h$ of the convex functions $g$ and $h$ at $x$ respectively.