- Source: Quasi-relative interior
In topology, a branch of mathematics, the quasi-relative interior of a subset of a vector space is a refinement of the concept of the interior. Formally, if
X
{\displaystyle X}
is a linear space then the quasi-relative interior of
A
⊆
X
{\displaystyle A\subseteq X}
is
qri
(
A
)
:=
{
x
∈
A
:
c
o
n
e
¯
(
A
−
x
)
is a linear subspace
}
{\displaystyle \operatorname {qri} (A):=\left\{x\in A:\operatorname {\overline {cone}} (A-x){\text{ is a linear subspace}}\right\}}
where
c
o
n
e
¯
(
⋅
)
{\displaystyle \operatorname {\overline {cone}} (\cdot )}
denotes the closure of the conic hull.
Let
X
{\displaystyle X}
be a normed vector space. If
C
⊆
X
{\displaystyle C\subseteq X}
is a convex finite-dimensional set then
qri
(
C
)
=
ri
(
C
)
{\displaystyle \operatorname {qri} (C)=\operatorname {ri} (C)}
such that
ri
{\displaystyle \operatorname {ri} }
is the relative interior.
See also
Interior (topology) – Largest open subset of some given set
Relative interior – Generalization of topological interior
Algebraic interior – Generalization of topological interior
References
Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.