Research Repository

# Compensated convex transforms and geometric singularity extraction from semiconvex functions

## Authors

KEWEI ZHANG Kewei.Zhang@nottingham.ac.uk
Professor of Mathematical Analysis

Elaine Crooks

Antonio Orlando

### Abstract

The 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.

### Citation

Zhang, K., Crooks, E., & Orlando, A. (in press). Compensated convex transforms and geometric singularity extraction from semiconvex functions. https://doi.org/10.1360/N012015-00339

Journal Article Type Article Mar 24, 2016 Apr 25, 2016 Aug 11, 2017 Aug 11, 2017 Scientia Sinica Mathematica 1674-7216 Peer Reviewed 46 5 https://doi.org/10.1360/N012015-00339 Compensated convex transforms, ridge transform, valley transform, edge transform, convex function, semiconvex function, semiconcave function, linear modulus, general modulus, DC-functions, singularity extraction, minimal bounding sphere, local approxima http://eprints.nottingham.ac.uk/id/eprint/44857 http://engine.scichina.com/publisher/scp/journal/SSM/46/5/10.1360/N012015-00339?slug=full text Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf A Chinese version of the material has been published in Zhang, Kewei, Crooks, Elaine and Orlando, Antonio, Compensated convex transforms and geometric singularity extraction from semiconvex functions (in Chinese), Sci. Sin. Math., 46 (2016) 747-768, doi: 10.1360/N012015-00339

#### Files

ZOC-semiconvex Zhang.pdf (300 Kb)
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