<s>
If	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
then	O
a	O
subset	O
of	O
is	O
bornivorous	B-Algorithm
if	O
it	O
is	O
bornivorous	B-Algorithm
with	O
respect	O
to	O
the	O
von-Neumann	B-Algorithm
bornology	I-Algorithm
of	O
.	O
</s>
<s>
Bornivorous	B-Algorithm
sets	I-Algorithm
play	O
an	O
important	O
role	O
in	O
the	O
definitions	O
of	O
many	O
classes	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
particularly	O
bornological	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
An	O
absorbing	B-Algorithm
disk	O
in	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
bornivorous	B-Algorithm
if	O
and	O
only	O
if	O
its	O
Minkowski	O
functional	O
is	O
locally	O
bounded	B-Algorithm
(	O
i.e.	O
</s>
<s>
maps	O
bounded	B-Algorithm
sets	I-Algorithm
to	O
bounded	B-Algorithm
sets	I-Algorithm
)	O
.	O
</s>
<s>
A	O
linear	O
map	O
between	O
two	O
TVSs	O
is	O
called	O
if	O
it	O
maps	O
Banach	B-Algorithm
disks	I-Algorithm
to	O
bounded	B-Algorithm
disks	O
.	O
</s>
<s>
A	O
disk	O
in	O
is	O
called	O
if	O
it	O
absorbs	B-Algorithm
every	O
Banach	B-Algorithm
disk	I-Algorithm
.	O
</s>
<s>
An	O
absorbing	B-Algorithm
disk	O
in	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
infrabornivorous	B-Algorithm
if	O
and	O
only	O
if	O
its	O
Minkowski	O
functional	O
is	O
infrabounded	O
.	O
</s>
<s>
A	O
disk	O
in	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
infrabornivorous	B-Algorithm
if	O
and	O
only	O
if	O
it	O
absorbs	B-Algorithm
all	O
compact	O
disks	O
(	O
that	O
is	O
,	O
if	O
it	O
is	O
""	O
)	O
.	O
</s>
<s>
Every	O
bornivorous	B-Algorithm
and	O
infrabornivorous	B-Algorithm
subset	O
of	O
a	O
TVS	O
is	O
absorbing	B-Algorithm
.	O
</s>
<s>
In	O
a	O
pseudometrizable	B-Algorithm
TVS	I-Algorithm
,	O
every	O
bornivore	B-Algorithm
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
Two	O
TVS	O
topologies	O
on	O
the	O
same	O
vector	O
space	O
have	O
that	O
same	O
bounded	B-Algorithm
subsets	O
if	O
and	O
only	O
if	O
they	O
have	O
the	O
same	O
bornivores	B-Algorithm
.	O
</s>
<s>
Suppose	O
is	O
a	O
vector	O
subspace	O
of	O
finite	O
codimension	O
in	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
and	O
If	O
is	O
a	O
barrel	O
(	O
resp	O
.	O
</s>
<s>
bornivorous	B-Algorithm
barrel	O
,	O
bornivorous	B-Algorithm
disk	O
)	O
in	O
then	O
there	O
exists	O
a	O
barrel	O
(	O
resp	O
.	O
</s>
<s>
Every	O
neighborhood	O
of	O
the	O
origin	O
in	O
a	O
TVS	O
is	O
bornivorous	B-Algorithm
.	O
</s>
<s>
The	O
convex	O
hull	O
,	O
closed	O
convex	O
hull	O
,	O
and	O
balanced	O
hull	O
of	O
a	O
bornivorous	B-Algorithm
set	I-Algorithm
is	O
again	O
bornivorous	B-Algorithm
.	O
</s>
<s>
The	O
preimage	O
of	O
a	O
bornivore	B-Algorithm
under	O
a	O
bounded	B-Algorithm
linear	O
map	O
is	O
a	O
bornivore	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
TVS	O
in	O
which	O
every	O
bounded	B-Algorithm
subset	I-Algorithm
is	O
contained	O
in	O
a	O
finite	O
dimensional	O
vector	O
subspace	O
,	O
then	O
every	O
absorbing	B-Algorithm
set	I-Algorithm
is	O
a	O
bornivore	B-Algorithm
.	O
</s>
<s>
If	O
is	O
the	O
balanced	O
hull	O
of	O
the	O
closed	O
line	O
segment	O
between	O
and	O
then	O
is	O
not	O
bornivorous	B-Algorithm
but	O
the	O
convex	O
hull	O
of	O
is	O
bornivorous	B-Algorithm
.	O
</s>
<s>
If	O
is	O
the	O
closed	O
and	O
"	O
filled	O
"	O
triangle	O
with	O
vertices	O
and	O
then	O
is	O
a	O
convex	O
set	O
that	O
is	O
not	O
bornivorous	B-Algorithm
but	O
its	O
balanced	O
hull	O
is	O
bornivorous	B-Algorithm
.	O
</s>
