DOI: 10.68381/jca22006 ISSN: 0944-6532
Extension of Continuous Convex Functions from Subspaces II
Carlo Alberto De Bernardi, Libor Veselý
Given
Y
Y
a subspace of a topological vector space
X
X
, and an open convex set
0\in A\subset X
0
∈
A
⊂
X
, we say that the couple
(X,Y)
(
X
,
Y
)
has the
\mathrm{CE}(A)
C
E
(
A
)
-property if each continuous convex function on
A\cap Y
A
∩
Y
admits a continuous convex extension defined on
A
A
. Using results from our previous paper, we study for given
A
A
the relation between the
\mathrm{CE}(A)
C
E
(
A
)
-property and the
\mathrm{CE}(X)
C
E
(
X
)
-property. As a corollary we obtain that
(X,Y)
(
X
,
Y
)
has the
\mathrm{CE}(A)
C
E
(
A
)
-property for each
A
A
, provided
(X,Y)
(
X
,
Y
)
has the
\mathrm{CE}(X)
C
E
(
X
)
-property and
Y
Y
is “conditionally separable”. This applies, for instance, if
X
X
is locally convex and conditionally separable. Other results concern either the
\mathrm{CE}(A)
C
E
(
A
)
-property for sets
A
A
of special forms, or the
\mathrm{CE}(A)
C
E
(
A
)
-property for each
A
A
where
X
X
is a normed space with
X/Y
X
/
Y
separable. In the last section, we point out connections between the
\mathrm{CE}(X)
C
E
(
X
)
-property and extendability of certain continuous linear operators. This easily yields a generalization of an extension theorem of Rosenthal, and another result of the same type.