2025/09/16 by Faruk Alpay, Hamdi Alakkad, Alpay, Faruk +1
Computer Science · Mathematics · #46B20 #47H09 #47H10 #65J15 #Advanced Banach Space Theory #D.3.4 #F.2.1 #FOS: Computer and information sciences #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Optimization and Variational Analysis #Programming Languages (cs.PL)
paper · pdf · doi:10.48550/arxiv.2509.13121
openalex publication_date 2025/09/16 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
Fixed point theory studies conditions under which nonexpansive maps on Banach spaces have fixed points. This paper examines the open question of whether every reflexive Banach space has the fixed point property. After surveying classical results, we propose a quantitative framework based on diametral l1 pressure and weighted selection functionals, which measure how much an orbit hull of a fixed point free nonexpansive map can collapse. We prove that if either invariant is uniformly positive, then the space must contain a copy of l1 and thus cannot be reflexive. We present finite dimensional certificates, positive and negative examples, and an x86-64 routine that computes mutual coherence and a lower bound for the pressure. The paper clarifies why existing approaches fail and outlines open problems and ethical considerations.