On Abstract Convexity with Respect to a Certain Quadratic Coupling Function: Subsets

Jakub Maksymiuk

doi:10.68381/jca34012

Abstract

We investigate abstract convexity generated by the coupling function

\varphi(x,A)=|A[x]^2|,\quad x\in \mathbb{R}^n,\ A\in \mathcal{L}^2_{sym}(\mathbb{R}^n,\mathbb{R}).

We study the

\varphi

-convexity of four classes of sets: subsets of

\mathbb{R}^n

, subsets of

\mathcal{L}

, epigraphic subsets of

\mathbb{R}^n\times \mathbb{R}

and epigraphic subsets of

\mathcal{L}\times\mathbb{R}

. We show that all families consist of closed, star-shaped sets and subsets of

\mathcal{L}

and

\mathcal{L}\times\mathbb{R}

are convex. For epigraphic subsets of

X\times\mathbb{R}

, we derive a condition partially characterizing them in analogy with classical convexity, where line segments are replaced by parabolas. In the one-dimensional case we obtain a complete characterization

Author information

Contact details are reproduced from the original publication and may be historical.

Jakub Maksymiuk

Institute of Applied Mathematics, Faculty of Technical Physics and Applied Mathematics, Gdańsk University of Technology, Poland

jakub.maksymiuk@pg.edu.pl

Keywords

Abstract convexity
coupling function
bilinear maps
quadratic functions

Mathematics Subject Classification

52A01, 52A30, 15A63

References

J. Appell, P. P. Zabrejko: Nonlinear Superposition Operators, Cambridge University Press, Cambridge (1990).https://doi.org/10.1017/CBO9780511897450 Google Scholar
G. Barletta, A. Cianchi: Dirichlet problems for fully anisotropic elliptic equations, Proc. Royal Soc. Edinburgh 147/1 (2017) 25–60.https://doi.org/10.1017/S0308210516000020 Google Scholar
M. Chmara, J. Maksymiuk: Anisotropic Orlicz-Sobolev spaces of vector valued functions and Lagrange equations, J. Math. Anal. Appl. 456/1 (2017) 457–475.https://doi.org/10.1016/j.jmaa.2017.07.032 Google Scholar
P. Clément, B. de Pagter, G. Sweers, F. de Thélin: Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces, Mediterr. J. Math. 1/3 (2004) 241–267.https://doi.org/10.1007/s00009-004-0014-6 Google Scholar
R. M. Dudley, R. Norvaiša: Concrete Functional Calculus, Springer, Berlin (2011).https://doi.org/10.1007/978-1-4419-6950-7 Google Scholar
J.-B. Hiriart-Urruty, C. Lemaréchal: Fundamentals of Convex Analysis, Springer, New York (2001).https://doi.org/10.1007/978-3-642-56468-0 Google Scholar
M. A. Krasnoselski, Ja. B. Rutickii: Convex Functions and Orlicz Spaces, Noordhoff, Groningen (1961).Google Scholar
J. Maksymiuk: On the Fenchel-Moreau conjugate of G-function and the second derivative of the modular in anisotropic Orlicz spaces, Calc. Var. Partial Diff. Equations 63/2 (2024) 1–36.https://doi.org/10.1007/s00526-023-02655-8 Google Scholar
D. E. Pallaschke, S. Rolewicz: Foundations of Mathematical Optimization: Convex Analysis without Linearity, Mathematics and its Applications Vol. 388, Kluwer-Springer, Dordrecht (2013).Google Scholar
A. M. Rubinov: Abstract Convexity and Global Optimization, Nonconvex Optimization and its Applications Vol. 44, Kluwer-Springer, Dordrecht (2000).https://doi.org/10.1007/978-1-4757-3200-9 Google Scholar
I. Singer: Abstract Convex Analysis, John Wiley & Sons, New York (1997).Google Scholar

Suggested citation

[1] J. Appell, P. P. Zabrejko: Nonlinear Superposition Operators, Cambridge University Press, Cambridge (1990).https://doi.org/10.1017/CBO9780511897450 Google Scholar

[2] G. Barletta, A. Cianchi: Dirichlet problems for fully anisotropic elliptic equations, Proc. Royal Soc. Edinburgh 147/1 (2017) 25–60.https://doi.org/10.1017/S0308210516000020 Google Scholar

[3] M. Chmara, J. Maksymiuk: Anisotropic Orlicz-Sobolev spaces of vector valued functions and Lagrange equations, J. Math. Anal. Appl. 456/1 (2017) 457–475.https://doi.org/10.1016/j.jmaa.2017.07.032 Google Scholar

[4] P. Clément, B. de Pagter, G. Sweers, F. de Thélin: Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces, Mediterr. J. Math. 1/3 (2004) 241–267.https://doi.org/10.1007/s00009-004-0014-6 Google Scholar

[5] R. M. Dudley, R. Norvaiša: Concrete Functional Calculus, Springer, Berlin (2011).https://doi.org/10.1007/978-1-4419-6950-7 Google Scholar

[6] J.-B. Hiriart-Urruty, C. Lemaréchal: Fundamentals of Convex Analysis, Springer, New York (2001).https://doi.org/10.1007/978-3-642-56468-0 Google Scholar

[7] M. A. Krasnoselski, Ja. B. Rutickii: Convex Functions and Orlicz Spaces, Noordhoff, Groningen (1961).Google Scholar

[8] J. Maksymiuk: On the Fenchel-Moreau conjugate of G-function and the second derivative of the modular in anisotropic Orlicz spaces, Calc. Var. Partial Diff. Equations 63/2 (2024) 1–36.https://doi.org/10.1007/s00526-023-02655-8 Google Scholar

[9] D. E. Pallaschke, S. Rolewicz: Foundations of Mathematical Optimization: Convex Analysis without Linearity, Mathematics and its Applications Vol. 388, Kluwer-Springer, Dordrecht (2013).Google Scholar

[10] A. M. Rubinov: Abstract Convexity and Global Optimization, Nonconvex Optimization and its Applications Vol. 44, Kluwer-Springer, Dordrecht (2000).https://doi.org/10.1007/978-1-4757-3200-9 Google Scholar

[11] I. Singer: Abstract Convex Analysis, John Wiley & Sons, New York (1997).Google Scholar