From Ham-Sandwich to Centerpoints: Semialgebraic Algorithms for Cutting Polytopal Measures
Pith reviewed 2026-07-03 01:26 UTC · model grok-4.3
The pith
The cap-volume function of polytopal measures is piecewise rational, enabling semialgebraic algorithms for ham-sandwich and centerpoint theorems.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The cap-volume function of a polytopal measure is piecewise rational on a natural decomposition of the space of oriented hyperplanes. This observation recasts prescribed-proportion cutting problems as semialgebraic feasibility problems. For fixed ambient dimension this yields polynomial-time algorithms to decide existence of cuts, describe the full solution set, and sample or enumerate solutions. The framework extends to the center transversal theorem, showing spaces of deep affine flats are semialgebraic, and the set of centerpoints of a convex polytope coincides with its floating body at level 1/(d+1).
What carries the argument
The piecewise rational cap-volume function defined on the space of oriented hyperplanes, which reduces cutting problems to semialgebraic feasibility.
If this is right
- Polynomial-time algorithms decide existence of ham-sandwich cuts and centerpoints in fixed dimension.
- The complete solution set of such cuts can be described explicitly as a semialgebraic set.
- Solutions can be sampled or enumerated in polynomial time when dimension is fixed.
- Spaces of deep affine flats for the center transversal theorem are semialgebraic.
- Centerpoints of any convex polytope equal its floating body at level 1/(d+1).
Where Pith is reading between the lines
- The algebraic description may allow exact rather than approximate computation of these partitions in low-dimensional instances that arise in practice.
- Similar piecewise-rational decompositions could be sought for other measure-partitioning results in discrete geometry.
- The explicit floating-body characterization of centerpoints supplies a new semialgebraic representation usable in optimization over convex bodies.
Load-bearing premise
The cap-volume function of a polytopal measure is piecewise rational on a natural decomposition of the space of oriented hyperplanes.
What would settle it
A concrete polytopal measure whose cap-volume function fails to be piecewise rational on every natural decomposition of oriented hyperplanes, or a convex polytope whose set of centerpoints differs from its floating body at level 1/(d+1).
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to design exact algorithms for the ham-sandwich and centerpoint theorems for polytopal measures. The key observation is that the cap-volume function of such a measure is piecewise rational on a natural decomposition of the space of oriented hyperplanes. This allows recasting prescribed-proportion cutting problems as semialgebraic feasibility problems, yielding polynomial-time algorithms in fixed ambient dimension to decide existence of cuts, describe the full solution set, and sample or enumerate solutions. The framework extends to the center transversal theorem by showing that spaces of deep affine flats are semialgebraic (which holds for centerpoints), and the set of centerpoints of a convex polytope is shown to coincide with its floating body at level 1/(d+1).
Significance. If the piecewise-rationality observation holds, the results would provide a useful algorithmic framework in computational geometry for exact solutions to classical measure-partitioning theorems, including polynomial-time procedures in fixed dimension and a semialgebraic description via floating bodies.
major comments (1)
- Abstract: the central algorithmic claims rest on the stated piecewise-rationality property of the cap-volume function, yet the abstract supplies no derivation, proof sketch, or verification. This property is load-bearing for the semialgebraic recasting and all complexity conclusions, so soundness cannot be assessed from the given text.
Simulated Author's Rebuttal
We thank the referee for their review. We address the single major comment below.
read point-by-point responses
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Referee: [—] Abstract: the central algorithmic claims rest on the stated piecewise-rationality property of the cap-volume function, yet the abstract supplies no derivation, proof sketch, or verification. This property is load-bearing for the semialgebraic recasting and all complexity conclusions, so soundness cannot be assessed from the given text.
Authors: Abstracts are concise summaries and do not contain derivations; the piecewise-rationality of the cap-volume function is derived in full in Section 3 of the manuscript by considering the arrangement of hyperplanes induced by the supporting hyperplanes of the polytopes in the measure. On each open cell of this arrangement the cap volume is a rational function of the hyperplane parameters, which directly yields the semialgebraic formulation used for the algorithmic results. The full text therefore supplies the required verification. revision: no
Circularity Check
No significant circularity identified
full rationale
The provided abstract states the piecewise-rational property of the cap-volume function as an independent key observation on a natural decomposition of oriented hyperplanes, which then enables recasting cutting problems as semialgebraic feasibility. No equations, self-citations, fitted parameters, or derivations are present that reduce any claim (such as the polynomial-time algorithms or the floating-body coincidence) back to its own inputs by construction. The derivation chain is therefore self-contained against the stated observation.
discussion (0)
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