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

We investigate abstract convexity generated by the coupling function φ(x,A)=∣A[x]2∣,x∈Rn, A∈Lsym2(Rn,R). We study the φ-convexity of four classes of sets: subsets of Rn, subsets of L, epigraphic subsets of Rn×R and epigraphic subsets of L×R. We show that all families consist of closed, star-shaped sets and subsets of L and L×R are convex. For epigraphic subsets of X×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