Integrated shape-sensitive functional metrics

TL;DR

This paper introduces a new integrated shape-sensitive metric bridging existing metrics and Lp distances, enabling finer analysis and practical computation in functional spaces.

Contribution

It develops an integrated ball (pseudo)metric that interpolates between a chosen metric and Lp distances, with applications to Hausdorff and Fréchet distances.

Findings

Provides a new class of shape-sensitive metrics.

Enables finer analysis in function spaces.

Facilitates practical computation through discrete approximations.

Abstract

This paper develops a new integrated ball (pseudo)metric which provides an intermediary between a chosen starting (pseudo)metric d and the L_p distance in general function spaces. Selecting d as the Hausdorff or Fr\'echet distances, we introduce integrated shape-sensitive versions of these supremum-based metrics. The new metrics allow for finer analyses in functional settings, not attainable applying the non-integrated versions directly. Moreover, convergent discrete approximations make computations feasible in practice.