Four classic problems

TL;DR

This paper surveys four foundational problems in geometry, providing an overview of their statements, significance, and related results to deepen understanding of these classic challenges.

Contribution

It offers a comprehensive review of four fundamental geometric problems, highlighting recent developments and open questions in the field.

Findings

Summarizes key results for each problem

Identifies open questions and research directions

Connects classical problems with modern geometric theory

Abstract

In this work we survey four classic problems: Borsuk's partition problem, Tarski's plank problem, the Kneser--Poulsen problem on the monotonicity of the union of balls under a contraction of their centers, and the Hadwiger--Levi problem on covering convex bodies by their smaller positively homothetic copies.