A note on Makeev's conjectures

TL;DR

This paper provides a counterexample to Makeev's Knaster-like conjecture on the sphere and offers positive solutions for specific cases of another Makeev conjecture regarding inscribing quadrangles.

Contribution

It presents a counterexample disproving Makeev's conjecture on $S^2$ and solves particular cases of a related inscribing quadrangle conjecture.

Findings

Counterexample to Makeev's Knaster-like conjecture on $S^2$

Positive solutions for specific cases of inscribing quadrangles

Advances understanding of geometric conjectures by Makeev

Abstract

A counterexample is given for the Knaster-like conjecture of Makeev for functions on $S^2$. Some particular cases of another conjecture of Makeev, on inscribing a quadrangle into a smooth simple closed curve, are solved positively.