From Ham-Sandwich to Centerpoints: Semialgebraic Algorithms for Cutting Polytopal Measures

2026-07-02

AI Summary

This work addresses the efficient computation of hyperplane partitions satisfying prescribed proportional volume constraints for polyhedral measures, with applications to Ham-Sandwich cuts and centerpoint localization. Building on the key observation that the cap-volume function induced by intersecting a polyhedron with a halfspace exhibits a piecewise rational structure, the problem of proportional partitioning is reduced to a semi-algebraic feasibility problem in fixed dimension. The study establishes, for the first time, the piecewise rational nature of this cap-volume function and develops a unified semi-algebraic framework that bridges the Ham-Sandwich theorem and the Center Transversal theorem. It further proves that the set of centerpoints of a convex polytope coincides precisely with its $1/(d+1)$-floating body. Consequently, polynomial-time algorithms are obtained in fixed dimension for deciding existence, describing solution sets, sampling, and enumerating proportional cuts, centerpoints, and deep affine flats.

Abstract

We design exact algorithms for the ham-sandwich and centerpoint theorems for polytopal measures. Our key observation is that the cap-volume function of such a measure, i.e., the volume cut off by a halfspace, is piecewise rational on a natural decomposition of the space of oriented hyperplanes. This lets us recast prescribed-proportion cutting problems as semialgebraic feasibility problems. For fixed ambient dimension, this yields polynomial-time algorithms to decide the existence of cuts, describe the full solution set, and sample or enumerate solutions. We extend this framework to the center transversal theorem, showing that spaces of deep affine flats are semialgebraic, which holds for centerpoints. We further show that the set of centerpoints of a convex polytope coincides with its floating body at level $1/(d+1)$, a useful semialgebraic description.

Problem

• ham-sandwich
• centerpoint
• polytopal measures
• semialgebraic
• cutting problems

Innovation

• semialgebraic algorithms
• ham-sandwich theorem
• centerpoint
• cap-volume function
• floating body