2023/06/03 by Abel Douzal, Douzal, Abel, Ferdinand Jacobé de Naurois +1
Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Geometric Topology (math.GT) #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2306.02041
openalex publication_date 2023/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Methods for measuring convexity defects of compacts in Rn abound. However, none of the those measures seems to take into account continuity. Continuity in convexity measure is essential for optimization, stability analysis, global optimality, convergence analysis, and accurate modelling as it ensures robustness and facilitates the development of efficient algorithms for solving convex optimization problems. This paper revisits the axioms underlying convexity measures by enriching them with a continuity hypothesis in Hausdorff's sense. Having provided the concept's theoretical grounds we state a theorem underlining the necessity of restricting ourselves to non-point compacts. We then construct a continuous convexity measure and compare it to existing measures. Importante note : This work is not a research article. It is an undergraduate project undertaken as part of a computer science course at École normale supérieure. It should therefore not be considered as a peer reviewed research paper.