On separability in discrete geometry

TL;DR

This paper surveys the development of research on separability and packings of convex bodies in discrete geometry, highlighting key results and open questions in the field.

Contribution

It provides a comprehensive overview of progress and unresolved problems related to separability in convex body arrangements in discrete geometry.

Findings

Summarizes key theorems and results in separability of convex bodies.

Identifies open problems and directions for future research.

Reviews progress on non-separable and totally separable packings.

Abstract

A problem of Erd\H{o}s (Amer. Math. Monthly 52: 494-498, 1945) and a theorem of Fejes T\'oth and Fejes T\'oth (Acta Math. Acad. Sci. Hungar. 24: 229-232, 1973) initiated the study of non-separable arrangements of convex bodies and the investigation of totally separable packings of convex bodies with both topics analyzing the concept of separability from the point view of discrete geometry. This article surveys the progress made on these and some closely related problems and highlights the relevant questions that have been left open.