A Hausdorff dimension for finite sets

The classical Hausdorff dimension, denoted dimH , of finite or countable sets is zero. We define an analog for finite sets, called finite Hausdorff dimension and denoted dimfH , which is non-trivial. It turns out that a finite bound for dimfH (F ) guarantees that every point of F has ”nearby” neighbors. This property is important for many computer algorithms of great practical value, that obtain solutions by finding nearest neighbors. We also define dimfB , an analog for finite sets of the classical box-counting dimension, and compute examples. The main result of the paper is a Convergence Theorem. It gives conditions under which, if Fn → X (convergence of compact subsets of Rn under the Hausdorff metric), then dimfH (Fn) → dimH (X).