DOI: 10.68381/jca34012 ISSN: 0944-6532
On Abstract Convexity with Respect to a Certain Quadratic Coupling Function: Subsets
Jakub Maksymiuk
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}).
φ
(
x
,
A
)
=
∣
A
[
x
]
2
∣
,
x
∈
R
n
,
A
∈
L
s
y
m
2
(
R
n
,
R
)
.
We study the
\varphi
φ
-convexity of four classes of sets: subsets of
\mathbb{R}^n
R
n
, subsets of
\mathcal{L}
L
, epigraphic subsets of
\mathbb{R}^n\times \mathbb{R}
R
n
×
R
and epigraphic subsets of
\mathcal{L}\times\mathbb{R}
L
×
R
. We show that all families consist of closed, star-shaped sets and subsets of
\mathcal{L}
L
and
\mathcal{L}\times\mathbb{R}
L
×
R
are convex. For epigraphic subsets of
X\times\mathbb{R}
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