Dimension preserving set-valued approximation and decomposition via metric sum
Ekta Agrawal, Saurabh Verma
TL;DR
The paper tackles the problem of approximating continuous set-valued functions with compact images without forcing convexity. It develops a fractal interpolation framework based on the metric sum via a Read-Bajraktarevič operator under a Rakotch contraction, proving the existence and uniqueness of a set-valued interpolant that interpolates data. It analyzes the dimensional properties of SVF graphs, establishing Falconer-type results for their distance and difference sets, and introduces a dimension-preserving decomposition: any continuous SVF can be expressed as a metric sum of components that preserve dimension. The results provide a toolkit for SVF approximation, analysis, and decomposition that avoids the undesired convexification of Minkowski sums, with potential impact on applications in control, differential inclusions, and fractal geometry.
Abstract
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.