From Ham-Sandwich to Centerpoints: Semialgebraic Algorithms for Cutting Polytopal Measures
arXiv:2607.02400
2026
Architecture
2 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper converts ham-sandwich and centerpoint existence statements into explicit semialgebraic computation by decomposing oriented-hyperplane space into cells on which polytopal cap volumes are rational functions. This enables exact or numerically controlled simultaneous mass-balance constraints rather than relying on stochastic gates and penalty tuning. The strongest ML transfer is a low-dimensional geometric router for mixture-of-experts or hierarchical clustering, where hyperplanes are solved to balance several token populations at prescribed proportions. A second, more exploratory transfer uses the floating-body characterization of centerpoints as a robust core estimator for embedding clouds.
Ideas from this paper
✗ Mechanism failed
2026
Replace a learned binary MoE gate with a hyperplane whose two sides contain prescribed proportions of several token populations simultaneously. In a low-dimensional routing projection, solve the cap-volume equations directly, producing deterministic per-population load control without an auxiliary load-balancing loss. Recursively applying the construction yields a balanced binary expert tree.
Useful7/10
Difficulty6/10
Novelty7/10
Unverified
2026
Construct a robust central region of each class or domain embedding cloud by intersecting halfspaces whose discarded cap mass is at most a prescribed fraction. Use this floating-body region to define prototypes or consistency targets, suppressing one-sided outliers without assuming Gaussian covariance structure. The centerpoint level 1/(d+1) provides a principled default depth parameter.
Useful5/10
Difficulty5/10
Novelty6/10