isn't this just because we're comparing n-dimensional objects by a 2-norm ? i.e. the dimension of the space grows but we're keeping the dimension of the norm fi

isn't this just because we're comparing n-dimensional objects by a 2-norm ? i.e. the dimension of the space grows but we're keeping the dimension of the norm fixed, but if we used the p-norm of the same dimension as the space, then maybe that would return intuitive results ?