<s>
In	O
the	O
analysis	B-General_Concept
of	I-General_Concept
algorithms	I-General_Concept
,	O
several	O
authors	O
have	O
studied	O
the	O
computation	O
of	O
the	O
volume	O
of	O
high-dimensional	O
convex	O
bodies	O
,	O
a	O
problem	O
that	O
can	O
also	O
be	O
used	O
to	O
model	O
many	O
other	O
problems	O
in	O
combinatorial	O
enumeration	O
.	O
</s>
<s>
It	O
is	O
known	O
that	O
,	O
in	O
this	O
model	O
,	O
no	O
deterministic	B-General_Concept
algorithm	I-General_Concept
can	O
achieve	O
an	O
accurate	O
approximation	O
,	O
and	O
even	O
for	O
an	O
explicit	O
listing	O
of	O
faces	O
or	O
vertices	O
the	O
problem	O
is	O
#P	O
-hard	O
.	O
</s>
<s>
However	O
,	O
a	O
joint	O
work	O
by	O
Martin	O
Dyer	O
,	O
Alan	O
M	O
.	O
Frieze	O
and	O
Ravindran	O
Kannan	O
provided	O
a	O
randomized	O
polynomial	B-Algorithm
time	I-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
for	O
the	O
problem	O
,	O
</s>
<s>
providing	O
a	O
sharp	O
contrast	O
between	O
the	O
capabilities	O
of	O
randomized	O
and	O
deterministic	B-General_Concept
algorithms	I-General_Concept
.	O
</s>
<s>
By	O
using	O
a	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
(	O
MCMC	O
)	O
method	O
,	O
it	O
is	O
possible	O
to	O
generate	O
points	O
that	O
are	O
nearly	O
uniformly	O
randomly	O
distributed	O
within	O
a	O
given	O
convex	O
body	O
.	O
</s>
<s>
By	O
using	O
rejection	B-Algorithm
sampling	I-Algorithm
,	O
it	O
is	O
possible	O
to	O
compare	O
the	O
volumes	O
of	O
two	O
convex	O
bodies	O
,	O
one	O
nested	O
within	O
another	O
,	O
when	O
their	O
volumes	O
are	O
within	O
a	O
small	O
factor	O
of	O
each	O
other	O
.	O
</s>
