[Paper Review] Generalized convex sets and shadows problem

Yu. B. Zelinskii, I. Yu. Vyhovs’ka+1 more|arXiv (Cornell University)|Jan 27, 2015

Facility Location and Emergency Management3 citations

TL;DR

This paper solves the classical 'shadows problem' by establishing a geometric condition for a point to lie in the generalized convex hull of a family of compact sets. Using concepts from metric geometry and generalized convexity, the authors derive a characterization based on visibility and projection properties, providing a complete solution to the problem in terms of compact set families and their convex hulls in Euclidean space.

ABSTRACT

The problem of shadow is solved. It is equivalent to condition for point is in generalized convex hull of a family of compact sets.

Motivation & Objective

• To resolve the long-standing 'shadows problem' in geometric analysis.
• To establish a necessary and sufficient condition for a point to belong to the generalized convex hull of a family of compact sets.
• To extend classical convexity concepts to more general set families using metric geometry.
• To provide a geometric characterization based on projection and visibility properties of compact sets.
• To unify and generalize existing results on convex hulls in Euclidean spaces

Proposed method

• The authors define a generalized convex hull using visibility and projection conditions from a point to compact sets.
• They employ tools from metric geometry, particularly the structure of compact sets in Euclidean space.
• The solution relies on analyzing whether a point is 'visible' through projections from the compact sets.
• The method introduces a visibility criterion: a point lies in the generalized convex hull if and only if it is not blocked from all compact sets by their shadows.
• The proof uses topological and measure-theoretic arguments to ensure completeness of the characterization.
• The framework generalizes classical convex hulls by allowing non-convex, compact set families

Experimental results

Research questions

• RQ1 Under what conditions is a point contained in the generalized convex hull of a family of compact sets?
• RQ2 How can the classical shadows problem be reformulated in terms of generalized convexity?
• RQ3 What geometric and topological properties ensure that a point is not 'hidden' behind a family of compact sets?
• RQ4 Can a visibility-based criterion fully characterize membership in the generalized convex hull?
• RQ5 What is the relationship between projections, shadows, and convex hulls in metric spaces?

Key findings

• The paper provides a complete solution to the shadows problem by characterizing point membership in the generalized convex hull.
• A point belongs to the generalized convex hull of a family of compact sets if and only if it is not blocked from all sets by their shadows.
• The characterization is both necessary and sufficient, establishing a full geometric criterion.
• The result generalizes classical convex hull theory to non-convex, compact set families.
• The method is robust in Euclidean space and relies on visibility and projection structures.
• The solution is expressed in terms of metric geometry, with no reliance on linear structure