Formalize point–interval closeness

Establish a formal measure of closeness between a single point and a closed interval, including cases in which the point lies outside the interval, for use in interval data analysis.

Background

The paper identifies a gap between mature point-to-point distance metrics and the tools available for interval-valued data. Although set-theoretic operations such as intersection and union can describe relationships between intervals, they do not provide a dedicated measure of how close an individual point is to an interval.

The unresolved issue is particularly relevant when the point is not contained in the interval. The paper proposes the point–interval closeness measure in Definition 3.1 to address this gap, but the explicitly stated open problem concerns the underlying formal quantification of point-to-interval closeness itself.

References

Nevertheless, a fundamental and still open question remains: how to formally quantify the closeness between a single point and an interval.

— Statistical Depth for Interval Data: A Closeness-Based Approach and Its Applications  (2609.24641 - Chen et al., 21 Sep 2026) in Section 3, subsection “A Closeness Measure Between Point and Interval”