On Pli\'{s} metric on the space of strictly convex compacta

TL;DR

This paper studies a specific metric on the space of convex compact sets in Euclidean space, demonstrating its completeness for strictly convex sets and exploring applications in set-valued analysis, including distance estimation between sets.

Contribution

The paper introduces and analyzes a particular metric on convex compacta, proving its completeness for strictly convex sets and applying it to set distance estimation problems.

Findings

The metric space of strictly convex compacta is complete under Pli's metric.

The paper provides bounds for the distance between convex sets using Pli's and Hausdorff metrics.

Applications to set-valued analysis are demonstrated with practical distance estimates.

Abstract

We consider a certain metric on the space of all convex compacta in $\R^{n}$, introduced by A. Pli\'s. The set of strictly convex compacta is a complete metric subspace of the metric space of convex compacta with respect to this metric. We present some applications of this metric to the problems of set-valued analysis, in particular we estimate the distance between two compact sets with respect to this metric and to the Hausdorff metric.