The normalized volume of a Kawamata log terminal pair at a point is the infimum of discrepancy-weighted volumes of valuations
centered at that point. For an 
-dimensional normal complex algebraic germ 
with rational Cartier canonical
divisor and zero boundary, assume 
is Kawamata log terminal. A real-valued additive
valuation 
centered at 
has ideals 
and volume
 |
where 
denotes module length. The normalized volume of
the germ is
 |
Here 
is log discrepancy and valuations of infinite
discrepancy contribute 
. The expression is invariant under positive rescaling
of 
.
The smooth-point value is 
.
