Dimension preserving set-valued approximation and decomposition via metric sum

In the literature, the Minkowski-sum and the metric-sum of compact sets are highlighted. While the first is associative, the latter is not. But the major drawback of the Minkowski combination is that, by increasing the number of summands, this leads to convexification. The present article is uncovered in two folds: The initial segment presents a novel approach to approximate a continuous set-valued function with compact images via a fractal approach using the metric linear combination of sets. The other segment contains the dimension analysis of the distance set of graph of set-valued function and solving the celebrated distance set conjecture. In the end, a decomposition of any continuous convex compact set-valued function is exhibited that preserves the Hausdorff dimension, so this will serve as a method for dealing with complicated set-valued functions.