mth roots of the identity operator and the geometry conjecture

2021/12/17 by Stephen Simons, Simons, Stephen
Computer Science · Mathematics · #46A22 #47H05 #47H10 #49J35 #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Optimization and Variational Analysis #Primary 46C05 #Secondary 46C07

paper · pdf · doi:10.48550/arxiv.2112.09805

openalex publication_date 2021/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• The difference vectors for convex sets and a resolution of the geometry conjecture
• Roots of the identity operator and proximal mappings: (classical and phantom) cycles and gap vectors
• On two recent conjectures in convex geometry
• Iterations of the projection body operator and a remark on Petty's conjectured projection inequality
• A geometric problem and the Hopf Lemma. I