<s>
A	O
zone	B-Application
diagram	I-Application
is	O
a	O
certain	O
geometric	O
object	O
which	O
a	O
variation	O
on	O
the	O
notion	O
of	O
Voronoi	B-Architecture
diagram	I-Architecture
.	O
</s>
<s>
When	O
we	O
speak	O
about	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
induced	O
by	O
these	O
sites	O
,	O
we	O
associate	O
to	O
the	O
site	O
the	O
set	O
of	O
all	O
points	O
in	O
the	O
plane	O
whose	O
distance	O
to	O
the	O
given	O
site	O
is	O
not	O
greater	O
to	O
their	O
distance	O
to	O
any	O
other	O
site	O
.	O
</s>
<s>
The	O
collection	O
of	O
these	O
regions	O
is	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
associated	O
with	O
these	O
sites	O
,	O
and	O
it	O
induces	O
a	O
decomposition	O
of	O
the	O
plane	O
into	O
regions	O
:	O
the	O
Voronoi	O
regions	O
(	O
Voronoi	B-Architecture
cells	I-Architecture
)	O
.	O
</s>
<s>
In	O
a	O
zone	B-Application
diagram	I-Application
the	O
region	O
associated	O
with	O
the	O
site	O
is	O
defined	O
a	O
little	O
bit	O
differently	O
:	O
instead	O
of	O
associating	O
it	O
the	O
set	O
of	O
all	O
points	O
whose	O
distance	O
to	O
is	O
not	O
greater	O
than	O
their	O
distance	O
to	O
the	O
other	O
sites	O
,	O
we	O
associate	O
to	O
the	O
set	O
of	O
all	O
points	O
in	O
the	O
plane	O
whose	O
distance	O
to	O
is	O
not	O
greater	O
than	O
their	O
distance	O
to	O
any	O
other	O
region	O
.	O
</s>
<s>
The	O
tuple	B-Application
is	O
the	O
zone	B-Application
diagram	I-Application
associated	O
with	O
the	O
sites	O
.	O
</s>
<s>
On	O
a	O
second	O
thought	O
,	O
we	O
see	O
that	O
actually	O
the	O
tuple	B-Application
is	O
a	O
solution	O
of	O
the	O
following	O
system	O
of	O
equations	O
:	O
</s>
<s>
Rigorously	O
,	O
a	O
zone	B-Application
diagram	I-Application
is	O
any	O
solution	O
of	O
this	O
system	O
,	O
if	O
such	O
a	O
solution	O
exists	O
.	O
</s>
<s>
In	O
some	O
settings	O
,	O
such	O
as	O
the	O
one	O
described	O
above	O
,	O
a	O
zone	B-Application
diagram	I-Application
can	O
be	O
interpreted	O
as	O
a	O
certain	O
equilibrium	O
between	O
mutually	O
hostile	O
kingdoms	O
,	O
.	O
</s>
<s>
Given	O
a	O
tuple	B-Application
of	O
nonempty	O
subsets	O
of	O
,	O
called	O
the	O
sites	O
,	O
a	O
zone	B-Application
diagram	I-Application
with	O
respect	O
to	O
this	O
tuple	B-Application
is	O
a	O
tuple	B-Application
of	O
subsets	O
of	O
such	O
that	O
for	O
all	O
the	O
following	O
equation	O
is	O
satisfied	O
:	O
</s>
<s>
The	O
system	O
of	O
equations	O
which	O
defines	O
the	O
zone	B-Application
diagram	I-Application
can	O
be	O
represented	O
as	O
a	O
fixed	O
point	O
of	O
a	O
function	O
defined	O
on	O
a	O
product	O
space	O
.	O
</s>
<s>
Then	O
is	O
a	O
zone	B-Application
diagram	I-Application
if	O
and	O
only	O
if	O
it	O
is	O
a	O
fixed	O
point	O
of	O
Dom	O
,	O
that	O
is	O
,	O
.	O
</s>
<s>
(	O
existence	O
and	O
uniqueness	O
of	O
the	O
zone	B-Application
diagram	I-Application
in	O
the	O
euclidean	O
plane	O
with	O
respect	O
to	O
finitely	O
many	O
point	O
sites	O
)	O
,	O
several	O
existence	O
or	O
existence	O
and	O
uniqueness	O
results	O
have	O
been	O
published	O
.	O
</s>
<s>
2	O
sites	O
(	O
existence	O
)	O
:	O
there	O
exists	O
at	O
least	O
one	O
zone	B-Application
diagram	I-Application
for	O
any	O
pair	O
of	O
arbitrary	O
sites	O
in	O
any	O
metric	O
space	O
.	O
</s>
<s>
More	O
than	O
2	O
sites	O
(	O
existence	O
)	O
:	O
there	O
exists	O
a	O
zone	B-Application
diagram	I-Application
of	O
finitely	O
many	O
compact	O
and	O
disjoint	O
sites	O
contained	O
in	O
the	O
interior	O
of	O
a	O
compact	O
and	O
convex	O
subset	O
of	O
a	O
uniformly	O
convex	O
space	O
.	O
</s>
<s>
More	O
than	O
2	O
sites	O
(	O
existence	O
and	O
uniqueness	O
)	O
:	O
there	O
exists	O
a	O
unique	O
zone	B-Application
diagram	I-Application
with	O
respect	O
to	O
any	O
collection	O
(	O
possibly	O
infinite	O
)	O
of	O
closed	O
and	O
positively	O
separated	O
sites	O
in	O
any	O
finite-dimensional	O
euclidean	O
space	O
.	O
</s>
<s>
In	O
addition	O
to	O
Voronoi	B-Architecture
diagrams	I-Architecture
,	O
zone	B-Application
diagrams	I-Application
are	O
closely	O
related	O
to	O
other	O
geometric	O
objects	O
such	O
as	O
double	O
zone	B-Application
diagrams	I-Application
,	O
trisectors	O
,	O
k-sectors	O
,	O
mollified	O
zone	B-Application
diagrams	I-Application
and	O
as	O
a	O
result	O
may	O
be	O
used	O
for	O
solving	O
problems	O
related	O
to	O
robot	O
motion	O
and	O
VLSI	O
design	O
,	O
.	O
</s>
