Journal of Convex Analysis 34 (2027), No. 1, 209--250
Copyright Heldermann Verlag 2027
On Abstract Convexity with Respect to a Certain Quadratic Coupling Function: Subsets
Jakub Maksymiuk
Institute of Applied Mathematics, Faculty of Technical Physics and Applied Mathematics, University of Technology, Gdansk, Poland
jakub.maksymiuk@pg.edu.pl
[Abstract-pdf]
We investigate abstract convexity generated by the coupling function \begin{equation*} \varphi(x,A)=|A[x]^2|,\quad x\in \mathbb{R}^n,\ A\in \mathcal{L}^2_{sym}(\mathbb{R}^n,\mathbb{R}). \end{equation*} 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.
Keywords: Abstract convexity, coupling function, bilinear maps, quadratic functions.
MSC: 52A01, 52A30, 15A63.
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