2025/05/20 by Xavier Goaoc, Goaoc, Xavier, Arnau Padrol +1
Engineering · Mathematics · #52C35: Arrangements of points #52C40: Oriented matroids #Advanced Materials and Mechanics #Combinatorics (math.CO) #Computational Geometry (cs.CG) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Mathematics and Applications #Structural Analysis and Optimization #flats #hyperplanes
paper · pdf · doi:10.48550/arxiv.2505.14189
openalex publication_date 2025/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let P be a finite full-dimensional point configuration in ℝd. We show that if a point configuration Q has the property that all finite chirotopes realizable by adding (generic) points to P are also realizable by adding points to Q, then P and Q are equal up to a direct affine transform. We also show that for any point configuration P and any ε>0, there is a finite, (generic) extension \widehat P of P with the following property: if another realization Q of the chirotope of P can be extended so as to realize the chirotope of \widehat P, then there exists a direct affine transform that maps each point of Q within distance ε of the corresponding point of P.