<s>
In	O
functional	B-Application
analysis	I-Application
,	O
a	O
branch	O
of	O
mathematics	O
,	O
the	O
distortion	B-Algorithm
problem	I-Algorithm
is	O
to	O
determine	O
by	O
how	O
much	O
one	O
can	O
distort	O
the	O
unit	O
sphere	O
in	O
a	O
given	O
Banach	O
space	O
using	O
an	O
equivalent	O
norm	O
.	O
</s>
<s>
Milman	O
showed	O
that	O
if	O
X	O
is	O
a	O
Banach	O
space	O
that	O
does	O
not	O
contain	O
an	O
isomorphic	O
copy	O
of	O
c0	O
or	O
ℓp	O
for	O
some	O
(	O
see	O
sequence	B-Algorithm
space	I-Algorithm
)	O
,	O
then	O
some	O
infinite-dimensional	O
subspace	O
of	O
X	O
is	O
distortable	O
.	O
</s>
<s>
So	O
the	O
distortion	B-Algorithm
problem	I-Algorithm
is	O
now	O
primarily	O
of	O
interest	O
on	O
the	O
spaces	O
ℓp	O
,	O
all	O
of	O
which	O
are	O
separable	O
and	O
uniform	O
convex	O
,	O
for	O
.	O
</s>
<s>
In	O
a	O
separable	O
Hilbert	O
space	O
,	O
the	O
distortion	B-Algorithm
problem	I-Algorithm
is	O
equivalent	O
to	O
the	O
question	O
of	O
whether	O
there	O
exist	O
subsets	O
of	O
the	O
unit	O
sphere	O
separated	O
by	O
a	O
positive	O
distance	O
and	O
yet	O
intersect	O
every	O
infinite-dimensional	O
closed	O
subspace	O
.	O
</s>
<s>
Unlike	O
many	O
properties	O
of	O
Banach	O
spaces	O
,	O
the	O
distortion	B-Algorithm
problem	I-Algorithm
seems	O
to	O
be	O
as	O
difficult	O
on	O
Hilbert	O
spaces	O
as	O
on	O
other	O
Banach	O
spaces	O
.	O
</s>
