Center of Mass in $N$ dimensions Does Center of Mass make sense in more than three dimensions? In the definition of the Center of Mass we have $dV$. Isn't the volume a three dimensional property? Or it is not and to define the center of mass for, say $4$ dimensions, we just have to calculate a four iterated integral instead of three?

It makes sense in any number of dimensions.  You are correct that for a continuous distribution you need one integral per dimension.  Volume is sometimes referred to as $4-$volume in $4$ dimensions.  $dV$ has the dimensionality of the space.
