Here is the problem: Given compact set $A\subset \mathbb{R}^{d}$, cover $\partial A$ by closed balls $\{B_{i,\varepsilon}\}_{i=1}^{n}$ , with minimum overlap. Can we express n as a factor of $\varepsilon$ and properties of A eg. $diam(A)$ i.e. $n=f(x,p(A))$?
To ensure coverage, more balls would have to be added for smaller $\varepsilon$.
Corollary: this will give us the total Newtonian capacity of the balls covering the boundary for fixed $\varepsilon$. And that is what I am looking for.
So if the boundary is a line of length L, we get $n=\frac{L}{2\varepsilon}$. So an estimate for higher dimensions is $n=\frac{vol(\partial A)}{vol(B(\varepsilon))}$ with $dim(\partial A)=dim (B(\varepsilon))$. I am looking for a formula of fixed $\varepsilon$ not it's limit to zero.
thanks
The asymptotic behavior of $n(\epsilon)$ can be somewhat complicated. It is described by means upper Minkowski / box dimension $$\overline{\dim}_M (\partial A)=\limsup_{\epsilon\to 0}\frac{\log n(\epsilon)}{\log(1/\epsilon)}$$ and its lower counterpart $$\underline{\dim}_M (\partial A)=\limsup_{\epsilon\to 0}\frac{\log n(\epsilon)}{\log(1/\epsilon)}$$ Unless $\partial A$ has intricate fractal structure that varies with scale (e.g., Cantor-type set where we sometimes remove middle $1/3$ and sometimes middle $1/4$), both of these dimensions coincide with the better-known Hausdorff dimension $\dim_H$. In general $$\dim_H \le \underline{\dim}_M \le \overline{\dim}_M$$ where both inequalities may be strict.
I do not know what you mean by "total Newtonian capacity of the boundary for fixed $\epsilon$." The Riesz capacities are related to the Hausdorff dimension: see the statement in formula (2.8) here and for the proof (and additional results), see Mattila's Geometry of Sets and Measures.
If $\partial A$ is a rectifiable surface with area $S$, then $n(\epsilon)$ is asymptotic to $ c S \epsilon^{1-n}$ where $c$ is an absolute constant that depends only on dimension. This quantity is also related to the volume of $\epsilon$-neighborhood of the set. For details, see the aforementioned book by Mattila.