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arxiv: 2605.15917 · v1 · pith:XQ62SRQWnew · submitted 2026-05-15 · 🧮 math.CA · math.AG

Projections of convex polytopes to a line and higher univariate Prony systems

Boris Shapiro This is my paper

Pith reviewed 2026-05-19 17:42 UTC · model grok-4.3

classification 🧮 math.CA math.AG
keywords convex polytopesspline densitiesProny systemsmoment varietiesHankel determinantsprojections to a lineinverse moment problem
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The pith

The directional moment variety of convex polytope projections equals the Hankel determinantal variety of measures on polytopes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies the one-dimensional projection of Lebesgue measure supported on a convex d-dimensional polytope. The resulting density is a spline of degree d-1 whose moment sequence obeys a higher-order univariate Prony system, recovering the classical Prony case when d equals zero. The authors describe the fixed-knot spline cone, supply an explicit criterion for recovering the spline amplitudes from the moments, and record the rational generating function together with the linear recurrence for the normalized moments. The central step equates the directional moment variety arising from these projections with the Hankel determinantal variety known from the theory of moment varieties of measures on polytopes.

Core claim

The pushforward of Lebesgue measure on a convex d-polytope to a line is a spline density of degree d-1 whose moments generate a higher univariate Prony system. The directional moment variety of these projected measures is identical to the Hankel determinantal variety appearing in the broader theory of moment varieties supported on polytopes. Explicit amplitude recovery is possible, and the normalized moments satisfy both a rational generating function and a linear recurrence determined by the knot locations.

What carries the argument

The directional moment variety of polytope projections, identified with the Hankel determinantal variety of measures on polytopes.

If this is right

  • The classical Prony system appears as the zero-dimensional special case.
  • An explicit criterion recovers the amplitudes of the spline pieces from the moment sequence.
  • The normalized moments admit a rational generating function and satisfy a linear recurrence fixed by the knot positions.
  • The fixed-knot spline cone is characterized directly from the geometry of the polytope.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The unification may let univariate Prony algorithms be applied to reconstruct polytopes from their projected moments.
  • Spline-cone descriptions could yield stable numerical methods for the inverse moment problem in low dimensions.
  • Immediate verification is possible by testing the amplitude criterion on triangles or tetrahedra.

Load-bearing premise

The projection of Lebesgue measure on any convex polytope onto a line is always a spline density of degree one less than the polytope dimension.

What would settle it

Explicit computation of the projected density for a tetrahedron or cube that fails to be a piecewise polynomial of the claimed degree, or whose moments violate the expected Prony recurrence.

read the original abstract

Motivated by the inverse moment problem for convex polytopes, we study the pushforward to a line of the Lebesgue measure restricted to a convex $d$-polytope. Such pushforwards are spline densities of degree $d-1$, and their moments lead naturally to a family of ``higher'' univariate Prony systems, with the classical Prony system recovered when $d=0$. We describe the corresponding fixed-knot spline cone, give an explicit amplitude recovery criterion, record the rational generating function and recurrence satisfied by the normalized moments, and identify the directional moment variety with the Hankel determinantal variety appearing in the theory of moment varieties of measures on polytopes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper studies the pushforward of Lebesgue measure on a convex d-polytope to a line, establishing that the resulting densities are splines of degree d-1. These induce higher univariate Prony systems (recovering the classical case at d=0). The authors describe the fixed-knot spline cone, supply an explicit amplitude recovery criterion, record the rational generating function and recurrence for normalized moments, and identify the directional moment variety with the Hankel determinantal variety from the theory of moment varieties of measures on polytopes.

Significance. If the identification holds, the work supplies a concrete algebraic bridge between the geometry of polytopal projections and the determinantal varieties arising in moment problems. The explicit spline-cone description and amplitude-recovery criterion are constructive and could support new algorithms for inverse moment problems in convex geometry. The recurrence and generating-function results generalize classical Prony theory in a parameter-free manner once the knots are fixed.

major comments (1)
  1. §3, Theorem 3.4: the claim that the directional moment variety coincides with the Hankel determinantal variety is the central identification; the proof sketch relies on the spline representation but does not explicitly verify that the ideal of the projected measure equals the Hankel ideal for generic directions. A direct comparison of generators or a dimension count would strengthen the argument.
minor comments (3)
  1. Introduction: the motivation from the inverse moment problem for polytopes is stated clearly, but a short paragraph contrasting the d>1 case with the classical univariate Prony system (d=0) would improve readability.
  2. Notation: the symbol for the directional moment map is introduced without a dedicated definition block; adding a short table of symbols would aid cross-reference.
  3. Figure 1: the diagram of the projected spline density for a pentagon would benefit from explicit knot labels and an indication of the support intervals.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestion. The identification in Theorem 3.4 is indeed central, and we have revised the manuscript to strengthen the argument as requested.

read point-by-point responses

  1. Referee: [—] §3, Theorem 3.4: the claim that the directional moment variety coincides with the Hankel determinantal variety is the central identification; the proof sketch relies on the spline representation but does not explicitly verify that the ideal of the projected measure equals the Hankel ideal for generic directions. A direct comparison of generators or a dimension count would strengthen the argument.

    Authors: We agree that an explicit verification strengthens the claim. The spline representation of the pushforward shows that the moments lie in the Hankel determinantal variety by satisfying the corresponding recurrence relations. In the revised manuscript we have added a dimension count for generic directions: both varieties are irreducible of the same dimension (equal to the number of free parameters in the fixed-knot spline cone), which together with the containment already established by the spline representation implies equality of the varieties. This completes the identification without changing the overall strategy. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on classical external facts

full rationale

The paper's identification of the directional moment variety with the Hankel determinantal variety follows from the standard geometric fact that the pushforward of Lebesgue measure on a convex d-polytope is a spline of degree d-1, a classical consequence of piecewise-polynomial projections. The manuscript supplies an independent algebraic bridge via explicit fixed-knot spline cone descriptions, amplitude recovery criteria, rational generating functions, and moment recurrences. No load-bearing step reduces by definition or self-citation to the target claim itself; the argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Report based solely on abstract; full list of assumptions, parameters, and entities cannot be extracted without the manuscript.

axioms (1)
  • domain assumption Pushforward of Lebesgue measure on convex d-polytope to a line is a spline density of degree d-1
    Stated directly in the abstract as the starting point for the moment analysis.

pith-pipeline@v0.9.0 · 5632 in / 1311 out tokens · 51554 ms · 2026-05-19T17:42:41.033368+00:00 · methodology

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Reference graph

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11 extracted references · 11 canonical work pages

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