The Hausdorff dimension of the hyperspace of compact sets

Abstract

Let \((X,\rho)\) be a separable metric space and let \((\mathcal{K}(X),\widetilde{\rho})\) denote the space of non-empty compact subsets of \(X\) with the Hausdorff metric. The purpose of this paper is to investigate the relationship of the Hausdorff dimension of a set \(E \subset X\) to that of \(\mathcal{K}(E) \subset \mathcal{K}(X)\).