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

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