Boundaries of logarithmic Voronoi cells can be transcendental

Yulia Alexandr

arXiv:2608.18434v1 Primary: math.ST Secondary: math.AG

Abstract

We show that logarithmic Voronoi boundaries for one-dimensional algebraic models in the probability simplex need not be algebraic. We give two explicit examples. For a union of two lines, the logarithmic Voronoi cell at a specific smooth rational point has a transcendental boundary point. For a cuspidal curve, the cell at the singular point contains a non-algebraic real-analytic boundary branch.

Disclosure

“this manuscript. I also thank Álvaro Ribot and Anna Seigal for useful discussions about Voronoi cells at singular points, which helped motivate my return to related questions in the logarithmic setting. The author acknowledges the use of ChatGPT for language editing and exploratory discussion. All mathematical results were developed and verified by the author. 10”
PDF page 10

Classification: Brainstorming or outlining
Multiplier: 2
Verified: 2026-08-22T18:31:47.302916Z

Structural counts

Pages 11 pdf

Theorems 3 source

Lemmas 1 source

Propositions 0 source

Corollaries 0 source

Definitions 0 source

Displayed equations 59 source

Bibliography entries 17 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.