Compensated convex transforms and geometric singularity extraction from semiconvex functionsTools Zhang, Kewei, Crooks, Elaine and Orlando, Antonio (2016) Compensated convex transforms and geometric singularity extraction from semiconvex functions. Scientia Sinica Mathematica, 46 (5). pp. 747-768. ISSN 1674-7216
Official URL: http://engine.scichina.com/publisher/scp/journal/SSM/46/5/10.1360/N012015-00339?slug=full text
AbstractThe upper and lower compensated convex transforms are `tight' one-sided approximations for a given function. We apply these transforms to the extraction of fine geometric singularities from general semiconvex/semiconcave functions and DC-functions in Rn (difference of convex functions). Well-known geometric examples of (locally) semiconcave functions include the Euclidean distance function and the Euclidean squared-distance function. For a locally semiconvex function f with general modulus, we show that `locally' a point is singular (a non-differentiable point) if and only if it is a scale 1-valley point, hence by using our method we can extract all fine singular points from a given semiconvex function. More precisely, if f is a semiconvex function with general modulus and x is a singular point, then locally the limit of the scaled valley transform exists at every point x and can be calculated as limλ→+∞λVλ(f)(x)=r2x/4, where rx is the radius of the minimal bounding sphere of the (Fréchet) subdifferential ∂−f(x) of the locally semiconvex f and Vλ(f)(x) is the valley transform at x. Thus the limit function V∞(f)(x):=limλ→+∞λVλ(f)(x)=r2x/4 provides a `scale 1-valley landscape function' of the singular set for a locally semiconvex function f. At the same time, the limit also provides an asymptotic expansion of the upper transform Cuλ(f)(x) when λ approaches +∞. For a locally semiconvex function f with linear modulus we show further that the limit of the gradient of the upper compensated convex transform limλ→+∞∇Cuλ(f)(x) exists and equals the centre of the minimal bounding sphere of ∂−f(x). We also show that for a DC-function f=g−h, the scale 1-edge transform, when λ→+∞, satisfies liminfλ→+∞λEλ(f)(x)≥(rg,x−rh,x)2/4, where rg,x and rh,x are the radii of the minimal bounding spheres of the subdifferentials ∂−g and ∂−h of the two convex functions g and h at x, respectively.
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