Hessian measures of semi-convex functions and applications to support measures of convex bodies

Colesanti, Andrea; Hug, Daniel

doi:10.1007/s002290050015

Cited by 29 publications

(29 citation statements)

References 35 publications

(49 reference statements)

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“…We will essentially use the following immediate consequence of [, Proposition 5.2. and 5.3]. Proposition Let

A \subset R^{d}

reach (A) > r > 0

and

a \in A

.…”

Section: Basic and Auxiliary Results On Sets Of Positive Reachmentioning

confidence: 99%

On the structure of sets with positive reach

Rataj

Zaj́ıček

2017

Mathematische Nachrichten

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We give a complete characterization of compact sets with positive reach (=proximally $C^1$ sets) in the plane and of one-dimensional sets with positive reach in ${\mathbb R}^d$. Further, we prove that if $\emptyset \neq A\subset{\mathbb R}^d$ is a set of positive reach of topological dimension $0< k \leq d$, then $A$ has its "$k$-dimensional regular part" $\emptyset \neq R \subset A$ which is a $k$-dimensional "uniform" $C^{1,1}$ manifold open in $A$ and $A\setminus R$ can be locally covered by finitely many $(k-1)$-dimensional DC surfaces. We also show that if $A \subset {\mathbb R}^d$ has positive reach, then $\partial A$ can be locally covered by finitely many semiconcave hypersurfaces

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“…We will essentially use the following immediate consequence of [, Proposition 5.2. and 5.3]. Proposition Let

A \subset R^{d}

reach (A) > r > 0

and

a \in A

.…”

Section: Basic and Auxiliary Results On Sets Of Positive Reachmentioning

confidence: 99%

On the structure of sets with positive reach

Rataj

Zaj́ıček

2017

Mathematische Nachrichten

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“…An important tool for proving the convergence results is the notion of semi-convex function (see for example [15] and references therein), which is reminded here.…”

Section: Semi-convex Functionsmentioning

confidence: 99%

Thresholding RULES and iterative shrinkage/thresholding algorithm: A convergence study

Kowalski

2014

2014 IEEE International Conference on Image Processing (ICIP)

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Imaging inverse problems can be formulated as an optimization problem and solved thanks to algorithms such as forward-backward or ISTA (Iterative Shrinkage/Thresholding Algorithm) for which non smooth functionals with sparsity constraints can be minimized efficiently. However, the soft thresholding operator involved in this algorithm leads to a biased estimation of large coefficients. That is why a step allowing to reduce this bias is introduced in practice. Indeed, in the statistical community, a large variety of thresholding operators have been studied to avoid the biased estimation of large coefficients; for instance, the non negative Garrote or the the SCAD thresholding. One can associate a non convex penalty to these operators. We study the convergence properties of ISTA, possibly relaxed, with any thresholding rule and show that they correspond to a semiconvex penalty. The effectiveness of this approach is illustrated on image inverse problems.

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“…In this context, the right notion of duality turned out to be the classical formation of the conjugate function. However, the theory developed in [6] does not seem to apply to the present situation.…”

Section: Resultsmentioning

confidence: 60%

“…the discussion in [14]. Still another kind of duality for Hessian measures of convex functions was discovered in [6]. In this context, the right notion of duality turned out to be the classical formation of the conjugate function.…”

Section: Resultsmentioning

confidence: 96%

See 1 more Smart Citation

Absolute Continuity for Curvature Measures of Convex Sets, III

Hug

2002

Advances in Mathematics

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This work is devoted to the investigation of the basic relationship between the geometric shape of a convex set and measure theoretic properties of the associated curvature and surface area measures. We study geometric consequences of and conditions for absolute continuity of curvature and surface area measures with respect to ðd À 1Þ-dimensional Hausdorff measure in Euclidean space R d : Our main results are two ''transfer principles'' which allow one to translate properties connected with the absolute continuity of the rth curvature measure of a convex body to dual properties related to the absolute continuity of the ðd À 1 À rÞth surface area measure of the polar body, and conversely. Applications are also considered. # 2002 Elsevier Science (USA)

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Hessian measures of semi-convex functions and applications to support measures of convex bodies

Cited by 29 publications

References 35 publications

On the structure of sets with positive reach

On the structure of sets with positive reach

Thresholding RULES and iterative shrinkage/thresholding algorithm: A convergence study

Absolute Continuity for Curvature Measures of Convex Sets, III

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