HESSIAN MEASURES OF CONVEX FUNCTIONS AND APPLICATIONS TO AREA MEASURES: Hessian measures of convex functions and applications to area measures
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| Title: | HESSIAN MEASURES OF CONVEX FUNCTIONS AND APPLICATIONS TO AREA MEASURES: Hessian measures of convex functions and applications to area measures |
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| Authors: | COLESANTI, ANDREA, D. HUG |
| Source: | Journal of the London Mathematical Society. 71:221-235 |
| Publisher Information: | Wiley, 2005. |
| Publication Year: | 2005 |
| Subject Terms: | Length, area, volume, other geometric measure theory, Abstract differentiation theory, differentiation of set functions, convex function, convex functions, \(j\)-extreme point, Convex functions and convex programs in convex geometry, Hausdorff and packing measures, Hessian measure, 0101 mathematics, Convex sets in \(n\) dimensions (including convex hypersurfaces), Radon-Nikodym derivative, 01 natural sciences |
| Description: | Let \(u:\Omega\to \mathbb R\) be a convex function on an open convex set of \(\mathbb R^d\). Its Hessian measures \(F_k(u;\cdot)\), \(k=0,1,\dots,d\), are defined through a local Steiner formula presenting the \(d\)-measure of the set \[ P_t(u;\omega):=\{x+ty: x\in \omega\text{ and }y\in\partial u(x)\} \] as a polynomial in \(t>0\) of degree \(d\). Here \(\omega\subset\mathbb R^d\) is a Borel set and \(\partial u(x)\) denotes the subdifferential of \(u\) at \(x\). For \(u\in C^2(\Omega)\) there is an alternative way to present \(F_k\)'s using the eigenvalues of the Hessian matrix \(D^2u\). The authors present a geometric description of \(\text{supp}\,F_k\) in terms of \(k\)-extreme points of \(u\), and investigate the Radon-Nikodym derivative and absolute continuity of \(F_k\) with respect to the \(d\)-measure of \(\mathbb R^d\). |
| Document Type: | Article |
| File Description: | application/xml |
| Language: | English |
| ISSN: | 1469-7750 0024-6107 |
| DOI: | 10.1112/s0024610704005915 |
| Access URL: | http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=280427 https://documat.unirioja.es/servlet/articulo?codigo=2206367 https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0024610704005915 https://dialnet.unirioja.es/servlet/articulo?codigo=2206367 https://academic.oup.com/jlms/article/71/1/221/932659 https://www.uni-due.de/~hm0045/Pub.collection/33.pdf https://hdl.handle.net/2158/251065 |
| Accession Number: | edsair.doi.dedup.....7438e90b3b9c8c8b036d79910f60f90d |
| Database: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | Let \(u:\Omega\to \mathbb R\) be a convex function on an open convex set of \(\mathbb R^d\). Its Hessian measures \(F_k(u;\cdot)\), \(k=0,1,\dots,d\), are defined through a local Steiner formula presenting the \(d\)-measure of the set \[ P_t(u;\omega):=\{x+ty: x\in \omega\text{ and }y\in\partial u(x)\} \] as a polynomial in \(t>0\) of degree \(d\). Here \(\omega\subset\mathbb R^d\) is a Borel set and \(\partial u(x)\) denotes the subdifferential of \(u\) at \(x\). For \(u\in C^2(\Omega)\) there is an alternative way to present \(F_k\)'s using the eigenvalues of the Hessian matrix \(D^2u\). The authors present a geometric description of \(\text{supp}\,F_k\) in terms of \(k\)-extreme points of \(u\), and investigate the Radon-Nikodym derivative and absolute continuity of \(F_k\) with respect to the \(d\)-measure of \(\mathbb R^d\). |
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| ISSN: | 14697750 00246107 |
| DOI: | 10.1112/s0024610704005915 |