$m$th roots of the identity operator and the geometry conjecture

TL;DR

This paper presents three novel proofs confirming the geometry conjecture related to cycles of projections in Hilbert spaces, utilizing minimax theorems, Fan's inequality, and properties of maximally monotone operators.

Contribution

It introduces three distinct new proofs of the geometry conjecture, expanding the theoretical understanding of projection cycles in Hilbert spaces.

Findings

The geometry conjecture is valid for cycles of projections.

Three different proof techniques are established.

The proofs connect minimax theorems, Fan's inequality, and monotone operator theory.

Abstract

In this paper, we give three different new proofs of the validity of the geometry conjecture about cycles of projections onto nonempty closed, convex subsets of a Hilbert space. The first uses a simple minimax theorem, which depends on the finite dimensional Hahn-Banach theorem. The second uses Fan's inequality, which has found many applications in optimization and mathematical economics. The third uses three results on maximally monotone operators on a Hilbert space.