An elementary proof of borsuk theorem

TL;DR

This paper offers a new, elementary proof of Borsuk's conjecture specifically for the two-dimensional case, simplifying the understanding of this geometric problem.

Contribution

It presents an alternative, elementary proof for Borsuk's theorem in the case of two dimensions, which was previously proved using more complex methods.

Findings

The conjecture holds true for d=2.

An elementary proof is provided for the 2D case.

The proof simplifies understanding of Borsuk's theorem in low dimensions.

Abstract

In 1933, Borsuk conjectured that any bounded d-dimensional set of nonzero diameter can be broken into d + 1 parts of smaller diameter. This conjecture was disproved for large enough d, though it is true for low dimensional cases. The paper provides an alternative proof for d = 2 case.