bibkey: delfsknebusch1984separation
authors: Hans Delfs and Manfred Knebusch
year: 1984
title: Separation, retractions and homotopy extension in semialgebraic spaces
doi: 10.2140/pjm.1984.114.47
url: https://msp.org/pjm/1984/114-1/p03.xhtml
claim: In the first barycentric subdivision of a simplex, an original vertex's open star is characterized by strict dominance of its parent coordinate, and its closure by the corresponding weak inequalities.
strata_touched:
  - D5/S3/Geometry/Gluing/BarycentricSubdivisionStar
license: citation-only
triage: anchor

Separation, retractions and homotopy extension in semialgebraic spaces

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Pacific Journal of Mathematics 114(1) (1984), 47–71. Publisher: https://msp.org/pjm/1984/114-1/p03.xhtml DOI: https://doi.org/10.2140/pjm.1984.114.47 Original paper: https://msp.org/pjm/1984/114-1/pjm-v114-n1-p03-p.pdf The star characterization is Lemma 2.6 on printed pages 56–57.

Star characterization and scope

For the first barycentric subdivision of a simplex, the open star of an original vertex consists of the points whose barycentric coordinate at that vertex strictly exceeds every other parent coordinate. Its closure is described by the corresponding weak inequalities. The open star includes the relative interiors of all incident subdivision faces, including the original singleton vertex. Removing zero coefficients retains the positive-support face chain in the strict case.

For a finite simplex, the closure of this open star is the union of all incident closed subdivision simplices: closure commutes with a finite union, and each simplex is the closure of its relative interior. This is the closed star used in the tetrahedral formalization. An affinely independent tetrahedron in a real normed vector space lies in its finite-dimensional affine span, so an infinite-dimensional ambient space does not enlarge the simplex case of the source result.

This existing result supplies the star-coordinate characterization used in CFMP geometric realization Lemma 70.3.