The Largest Eigenvalue of a Convex Function, Duality, and a Theorem of Slodkowski

First, we provide an exposition of a theorem due to Slodkowski regarding the largest “eigenvalue” of a convex function. In his work on the Dirichlet problem, Slodkowski introduces a generalized second-order derivative which for C 2 functions corresponds to the largest eigenvalue of the Hessian. The...

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Published in:	The Journal of geometric analysis Vol. 26; no. 4; pp. 3027 - 3055
Main Author:	Dellatorre, Matthew
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.10.2016 Springer Nature B.V
Subjects:	Abstract Harmonic Analysis Convex and Discrete Geometry Convexity Differential Geometry Dirichlet problem Dynamical Systems and Ergodic Theory Eigenvalues Fourier Analysis Geometry Global Analysis and Analysis on Manifolds Lower bounds Mathematics Mathematics and Statistics Theorems 52A41 35J60 Dualilty 26B25 Convex analysis Legendre transform estimates
ISSN:	1050-6926, 1559-002X
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Abstract	First, we provide an exposition of a theorem due to Slodkowski regarding the largest “eigenvalue” of a convex function. In his work on the Dirichlet problem, Slodkowski introduces a generalized second-order derivative which for C 2 functions corresponds to the largest eigenvalue of the Hessian. The theorem allows one to extend an a.e. lower bound on this largest “eigenvalue” to a bound holding everywhere. Via the Dirichlet duality theory of Harvey and Lawson, this result has been key to recent progress on the fully non-linear, elliptic Dirichlet problem. Second, using the Legendre–Fenchel transform we derive a dual characterization of this largest eigenvalue in terms of convexity of the conjugate function. This dual characterization offers further insight into the nature of this largest eigenvalue and allows for an alternative proof of a necessary bound for the theorem.
AbstractList	First, we provide an exposition of a theorem due to Slodkowski regarding the largest “eigenvalue” of a convex function. In his work on the Dirichlet problem, Slodkowski introduces a generalized second-order derivative which for C 2 functions corresponds to the largest eigenvalue of the Hessian. The theorem allows one to extend an a.e. lower bound on this largest “eigenvalue” to a bound holding everywhere. Via the Dirichlet duality theory of Harvey and Lawson, this result has been key to recent progress on the fully non-linear, elliptic Dirichlet problem. Second, using the Legendre–Fenchel transform we derive a dual characterization of this largest eigenvalue in terms of convexity of the conjugate function. This dual characterization offers further insight into the nature of this largest eigenvalue and allows for an alternative proof of a necessary bound for the theorem. First, we provide an exposition of a theorem due to Slodkowski regarding the largest “eigenvalue” of a convex function. In his work on the Dirichlet problem, Slodkowski introduces a generalized second-order derivative which for C 2 functions corresponds to the largest eigenvalue of the Hessian. The theorem allows one to extend an a.e. lower bound on this largest “eigenvalue” to a bound holding everywhere. Via the Dirichlet duality theory of Harvey and Lawson, this result has been key to recent progress on the fully non-linear, elliptic Dirichlet problem. Second, using the Legendre–Fenchel transform we derive a dual characterization of this largest eigenvalue in terms of convexity of the conjugate function. This dual characterization offers further insight into the nature of this largest eigenvalue and allows for an alternative proof of a necessary bound for the theorem.
Author	Dellatorre, Matthew
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Cites_doi	10.1002/cpa.20265 10.1515/9781400873173 10.1007/978-3-662-02796-7 10.1201/9781420022605
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References	Rogers (CR8) 1970 CR3 Hiriart-Urruty, Lemaráchal (CR6) 1993 Rockafellar (CR7) 1970 CR5 Slodkowski (CR9) 1984; 11 Animov (CR2) 2001 Harvey, Lawson (CR4) 2009; 62 Alexandrov (CR1) 1939; 37 CA Rogers (9660_CR8) 1970 AD Alexandrov (9660_CR1) 1939; 37 Y Animov (9660_CR2) 2001 FR Harvey (9660_CR4) 2009; 62 9660_CR3 Z Slodkowski (9660_CR9) 1984; 11 J-B Hiriart-Urruty (9660_CR6) 1993 9660_CR5 RT Rockafellar (9660_CR7) 1970
References_xml	– year: 2001 ident: CR2 publication-title: Differential Geometry and Topology of Curves – volume: 62 start-page: 396 year: 2009 end-page: 443 ident: CR4 article-title: Dirichlet duality and the non-linear Dirichlet problem publication-title: Commun. Pure Appl. Math. doi: 10.1002/cpa.20265 – ident: CR5 – year: 1993 ident: CR6 publication-title: Convex Analysis and Minimization Algorithms – ident: CR3 – year: 1970 ident: CR7 publication-title: Convex Analysis doi: 10.1515/9781400873173 – year: 1970 ident: CR8 publication-title: Hausdorff Measures – volume: 11 start-page: 303 issue: 2 year: 1984 end-page: 326 ident: CR9 article-title: The Bremermann–Dirichlet problem for q-Plurisubharmonic functions publication-title: Analli della Scuola Normale – volume: 37 start-page: 3 year: 1939 end-page: 35 ident: CR1 article-title: Almost everywhere existence of the second differential of a convex function and properties of convex surfaces connected with it (in Russian) publication-title: Leningrad State Univ. Ann. Math. – volume-title: Convex Analysis and Minimization Algorithms year: 1993 ident: 9660_CR6 doi: 10.1007/978-3-662-02796-7 – volume: 37 start-page: 3 year: 1939 ident: 9660_CR1 publication-title: Leningrad State Univ. Ann. Math. – volume: 62 start-page: 396 year: 2009 ident: 9660_CR4 publication-title: Commun. Pure Appl. Math. doi: 10.1002/cpa.20265 – volume-title: Hausdorff Measures year: 1970 ident: 9660_CR8 – volume-title: Differential Geometry and Topology of Curves year: 2001 ident: 9660_CR2 doi: 10.1201/9781420022605 – ident: 9660_CR5 – volume-title: Convex Analysis year: 1970 ident: 9660_CR7 doi: 10.1515/9781400873173 – volume: 11 start-page: 303 issue: 2 year: 1984 ident: 9660_CR9 publication-title: Analli della Scuola Normale – ident: 9660_CR3
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SubjectTerms	Abstract Harmonic Analysis Convex and Discrete Geometry Convexity Differential Geometry Dirichlet problem Dynamical Systems and Ergodic Theory Eigenvalues Fourier Analysis Geometry Global Analysis and Analysis on Manifolds Lower bounds Mathematics Mathematics and Statistics Theorems
Title	The Largest Eigenvalue of a Convex Function, Duality, and a Theorem of Slodkowski
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