Gaussian polyhedral partitions with prescribed masses and barycenters
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Can one partition Gaussian space into convex polyhedral cells with prescribed masses and projected barycenters? The coupling target assumes a centered finite-support random vector X is convexly dominated by ρG for a standard Gaussian G and 0<ρ<1.
Why it matters and what counts as progress
Why it matters: The partition lemma provides a geometric route from convex domination to exact mass and barycenter constraints.
What counts as progress: Prove the partition and barycenter constraints through a martingale coupling or extend the lemma beyond strict Gaussian domination.
How the problem developed
- 2025 · Partial result [1]
David P. Bourne, Thomas Gallouët, Quentin Mérigot, and Andrea Natale Established a convex-order characterization for discrete measures dominated by an absolutely continuous measure and a method for fitting Laguerre tessellations; this supplies a broader geometric tool behind the later coupling proposition.
- 11 May 2026 · Reported claim [2]
ChatGPT-5.5 Pro, with Dongming Merrick Hua, Antoine Song, and Stefan Tudose Appendix B states a Gaussian polyhedral partition proposition with prescribed cell masses and conditional barycenters; the authors note that the proposition also follows from the earlier Laguerre-tessellation theorem and provide their own sketch.
AI contribution
On Talagrand's Convexity Conjecture
Appendix B Proposition B.1 gives a convex polyhedral Gaussian partition with prescribed cell masses and projected barycenters when a centered finite-support X is convexly dominated by ρG for 0<ρ<1. The GPT proof is an alternative martingale-coupling step, not the paper’s main Talagrand theorem.
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