A quantitative program for Hadwiger's covering conjecture and Borsuk's partition conjecture

TL;DR

This paper introduces a unified quantitative approach to Hadwiger's covering conjecture and Borsuk's partition conjecture by encoding them into continuous functions on convex bodies and proposing a four-step strategy.

Contribution

It presents a novel framework encoding two longstanding conjectures into continuous functions and outlines a four-step program to advance their resolution.

Findings

Partial results obtained for the conjectures

A new encoding method for the conjectures

A proposed strategy for future research

Abstract

In this article we encode Hadwiger's covering conjecture and Borsuk's partition conjecture into continuous functions defined on the spaces of convex bodies, propose a four-step program to approach them, and obtain some partial results.