<s>
In	O
geometry	O
,	O
the	O
geometric	B-General_Concept
median	I-General_Concept
of	O
a	O
discrete	O
set	O
of	O
sample	O
points	O
in	O
a	O
Euclidean	O
space	O
is	O
the	O
point	O
minimizing	O
the	O
sum	O
of	O
distances	O
to	O
the	O
sample	O
points	O
.	O
</s>
<s>
It	O
is	O
also	O
known	O
as	O
the	O
1-median	B-General_Concept
,	O
spatial	O
median	O
,	O
Euclidean	O
minisum	O
point	O
,	O
or	O
Torricelli	O
point	O
.	O
</s>
<s>
The	O
geometric	B-General_Concept
median	I-General_Concept
is	O
an	O
important	O
estimator	O
of	O
location	O
in	O
statistics	O
,	O
where	O
it	O
is	O
also	O
known	O
as	O
the	O
L1	O
estimator	O
(	O
after	O
the	O
L1	O
norm	O
)	O
.	O
</s>
<s>
The	O
more	O
general	O
k-median	B-Algorithm
problem	I-Algorithm
asks	O
for	O
the	O
location	O
of	O
k	O
cluster	O
centers	O
minimizing	O
the	O
sum	O
of	O
distances	O
from	O
each	O
sample	O
point	O
to	O
its	O
nearest	O
center	O
.	O
</s>
<s>
If	O
the	O
point	O
is	O
generalized	O
into	O
a	O
line	O
or	O
a	O
curve	O
,	O
the	O
best-fitting	O
solution	O
is	O
found	O
via	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
.	O
</s>
<s>
The	O
geometric	B-General_Concept
median	I-General_Concept
may	O
in	O
turn	O
be	O
generalized	O
to	O
the	O
problem	O
of	O
minimizing	O
the	O
sum	O
of	O
weighted	O
distances	O
,	O
known	O
as	O
the	O
Weber	O
problem	O
after	O
Alfred	O
Weber	O
's	O
discussion	O
of	O
the	O
problem	O
in	O
his	O
1909	O
book	O
on	O
facility	O
location	O
.	O
</s>
<s>
Some	O
sources	O
instead	O
call	O
Weber	O
's	O
problem	O
the	O
Fermat	O
–	O
Weber	O
problem	O
,	O
but	O
others	O
use	O
this	O
name	O
for	O
the	O
unweighted	O
geometric	B-General_Concept
median	I-General_Concept
problem	O
.	O
</s>
<s>
provides	O
a	O
survey	O
of	O
the	O
geometric	B-General_Concept
median	I-General_Concept
problem	O
.	O
</s>
<s>
For	O
the	O
1-dimensional	O
case	O
,	O
the	O
geometric	B-General_Concept
median	I-General_Concept
coincides	O
with	O
the	O
median	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
univariate	B-General_Concept
median	O
also	O
minimizes	O
the	O
sum	O
of	O
distances	O
from	O
the	O
points	O
.	O
</s>
<s>
(	O
More	O
precisely	O
,	O
if	O
the	O
points	O
are	O
p1	O
,	O
…,	O
pn	O
,	O
in	O
that	O
order	O
,	O
the	O
geometric	B-General_Concept
median	I-General_Concept
is	O
the	O
middle	O
point	O
if	O
n	O
is	O
odd	O
,	O
but	O
is	O
not	O
uniquely	O
determined	O
if	O
n	O
is	O
even	O
,	O
when	O
it	O
can	O
be	O
any	O
point	O
in	O
the	O
line	O
segment	O
between	O
the	O
two	O
middling	O
points	O
and	O
.	O
)	O
</s>
<s>
The	O
geometric	B-General_Concept
median	I-General_Concept
is	O
unique	O
whenever	O
the	O
points	O
are	O
not	O
collinear	O
.	O
</s>
<s>
The	O
geometric	B-General_Concept
median	I-General_Concept
is	O
equivariant	O
for	O
Euclidean	O
similarity	O
transformations	O
,	O
including	O
translation	B-Algorithm
and	O
rotation	B-General_Concept
.	O
</s>
<s>
This	O
means	O
that	O
one	O
would	O
get	O
the	O
same	O
result	O
either	O
by	O
transforming	O
the	O
geometric	B-General_Concept
median	I-General_Concept
,	O
or	O
by	O
applying	O
the	O
same	O
transformation	O
to	O
the	O
sample	O
data	O
and	O
finding	O
the	O
geometric	B-General_Concept
median	I-General_Concept
of	O
the	O
transformed	O
data	O
.	O
</s>
<s>
This	O
property	O
follows	O
from	O
the	O
fact	O
that	O
the	O
geometric	B-General_Concept
median	I-General_Concept
is	O
defined	O
only	O
from	O
pairwise	O
distances	O
,	O
and	O
does	O
not	O
depend	O
on	O
the	O
system	O
of	O
orthogonal	O
Cartesian	O
coordinates	O
by	O
which	O
the	O
sample	O
data	O
is	O
represented	O
.	O
</s>
<s>
In	O
contrast	O
,	O
the	O
component-wise	O
median	O
for	O
a	O
multivariate	O
data	O
set	O
is	O
not	O
in	O
general	O
rotation	B-General_Concept
invariant	O
,	O
nor	O
is	O
it	O
independent	O
of	O
the	O
choice	O
of	O
coordinates	O
.	O
</s>
<s>
The	O
geometric	B-General_Concept
median	I-General_Concept
has	O
a	O
breakdown	O
point	O
of	O
0.5	O
.	O
</s>
<s>
For	O
3	O
(	O
non-collinear	O
)	O
points	O
,	O
if	O
any	O
angle	O
of	O
the	O
triangle	O
formed	O
by	O
those	O
points	O
is	O
120°	O
or	O
more	O
,	O
then	O
the	O
geometric	B-General_Concept
median	I-General_Concept
is	O
the	O
point	O
at	O
the	O
vertex	O
of	O
that	O
angle	O
.	O
</s>
<s>
If	O
all	O
the	O
angles	O
are	O
less	O
than	O
120°	O
,	O
the	O
geometric	B-General_Concept
median	I-General_Concept
is	O
the	O
point	O
inside	O
the	O
triangle	O
which	O
subtends	O
an	O
angle	O
of	O
120°	O
to	O
each	O
three	O
pairs	O
of	O
triangle	O
vertices	O
.	O
</s>
<s>
(	O
If	O
the	O
three	O
points	O
are	O
collinear	O
then	O
the	O
geometric	B-General_Concept
median	I-General_Concept
is	O
the	O
point	O
between	O
the	O
two	O
other	O
points	O
,	O
as	O
is	O
the	O
case	O
with	O
a	O
one-dimensional	O
median	O
.	O
)	O
</s>
<s>
For	O
4	O
coplanar	O
points	O
,	O
if	O
one	O
of	O
the	O
four	O
points	O
is	O
inside	O
the	O
triangle	O
formed	O
by	O
the	O
other	O
three	O
points	O
,	O
then	O
the	O
geometric	B-General_Concept
median	I-General_Concept
is	O
that	O
point	O
.	O
</s>
<s>
Otherwise	O
,	O
the	O
four	O
points	O
form	O
a	O
convex	O
quadrilateral	O
and	O
the	O
geometric	B-General_Concept
median	I-General_Concept
is	O
the	O
crossing	O
point	O
of	O
the	O
diagonals	O
of	O
the	O
quadrilateral	O
.	O
</s>
<s>
The	O
geometric	B-General_Concept
median	I-General_Concept
of	O
four	O
coplanar	O
points	O
is	O
the	O
same	O
as	O
the	O
unique	O
Radon	O
point	O
of	O
the	O
four	O
points	O
.	O
</s>
<s>
Despite	O
the	O
geometric	B-General_Concept
median	I-General_Concept
's	O
being	O
an	O
easy-to-understand	O
concept	O
,	O
computing	O
it	O
poses	O
a	O
challenge	O
.	O
</s>
<s>
The	O
centroid	O
or	O
center	O
of	O
mass	O
,	O
defined	O
similarly	O
to	O
the	O
geometric	B-General_Concept
median	I-General_Concept
as	O
minimizing	O
the	O
sum	O
of	O
the	O
squares	O
of	O
the	O
distances	O
to	O
each	O
point	O
,	O
can	O
be	O
found	O
by	O
a	O
simple	O
formula	O
—	O
its	O
coordinates	O
are	O
the	O
averages	O
of	O
the	O
coordinates	O
of	O
the	O
points	O
—	O
but	O
it	O
has	O
been	O
shown	O
that	O
no	O
explicit	O
formula	O
,	O
nor	O
an	O
exact	O
algorithm	O
involving	O
only	O
arithmetic	O
operations	O
and	O
kth	O
roots	O
,	O
can	O
exist	O
in	O
general	O
for	O
the	O
geometric	B-General_Concept
median	I-General_Concept
.	O
</s>
<s>
However	O
,	O
it	O
is	O
straightforward	O
to	O
calculate	O
an	O
approximation	O
to	O
the	O
geometric	B-General_Concept
median	I-General_Concept
using	O
an	O
iterative	O
procedure	O
in	O
which	O
each	O
step	O
produces	O
a	O
more	O
accurate	O
approximation	O
.	O
</s>
<s>
One	O
common	O
approach	O
of	O
this	O
type	O
,	O
called	O
Weiszfeld	O
's	O
algorithm	O
after	O
the	O
work	O
of	O
Endre	O
Weiszfeld	O
,	O
is	O
a	O
form	O
of	O
iteratively	B-Algorithm
re-weighted	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
show	O
how	O
to	O
compute	O
the	O
geometric	B-General_Concept
median	I-General_Concept
to	O
arbitrary	O
precision	O
in	O
nearly	O
linear	O
time	O
.	O
</s>
<s>
If	O
y	O
is	O
distinct	O
from	O
all	O
the	O
given	O
points	O
,	O
xi	O
,	O
then	O
y	O
is	O
the	O
geometric	B-General_Concept
median	I-General_Concept
if	O
and	O
only	O
if	O
it	O
satisfies	O
:	O
</s>
<s>
In	O
general	O
,	O
y	O
is	O
the	O
geometric	B-General_Concept
median	I-General_Concept
if	O
and	O
only	O
if	O
there	O
are	O
vectors	O
ui	O
such	O
that	O
:	O
</s>
<s>
The	O
geometric	B-General_Concept
median	I-General_Concept
can	O
be	O
generalized	O
from	O
Euclidean	O
spaces	O
to	O
general	O
Riemannian	B-Architecture
manifolds	I-Architecture
(	O
and	O
even	O
metric	O
spaces	O
)	O
using	O
the	O
same	O
idea	O
which	O
is	O
used	O
to	O
define	O
the	O
Fréchet	O
mean	O
on	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
.	O
</s>
<s>
If	O
all	O
the	O
weights	O
are	O
equal	O
,	O
we	O
say	O
simply	O
that	O
is	O
the	O
geometric	B-General_Concept
median	I-General_Concept
.	O
</s>
