The Slide Dimension of Point Processes

We associate with any finite subset of a metric space an infinite sequence of scale invariant numbers ${\rho }_1,{\rho }_2,\dots$ derived from a variant of differential entropy called the genial entropy. As statistics for point processes, these numbers often appear to converge in simulations and we give examples where $1\mathrm{/}{\rho }_1$ converges to the Hausdorff dimension. We use the ${\rho }_n$ to define a new notion of dimension called the slide dimension for a special class of point processes on metric spaces. The slide calculus is developed to define ${\rho }_n$ and an explicit formula is derived for the calculation of ${\rho }_1$. For a uniform random variable X on $[0,1{]}^n$, evidence is given that ${\rho }_1(X)=1\mathrm{/}n$ and ${\rho }_2(X)=-{\pi }^2\mathrm{/}(6n^2)$ and simulations with a normal variable $Z$ suggest that ${\rho }_1(Z)=4\mathrm{/}\pi$ and ${\rho }_2(Z)=-1$. Some potential applications to spatial statistics are considered.