  We consider convolving a Gaussian of a varying scale epsilon against a Borel
measure mu on Euclidean delta-dimensional space. The Lq norm of the result is
differentiable in epsilon. We calculate this derivative and show how the upper
order of its growth relates to its lower Renyi dimension. We assume q is
strictly between 1 and infty and that mu is finite with compact support.
  Consider choosing a sequence epsilon_n of scales for the Gaussians. The
differences between the usual Lq norms at adjacent scales can be made to grow
more slowly than any positive power of n by setting the epsilon_n by a power
rule. The correct exponent in the power rule is determined by the lower Renyi
dimension.
  We calculate and find bounds on the derivative of the Gaussian kernel
versions of the correlation integral. We show that a Gaussian Kernel version of
the Renyi entropy sum in continuous.
