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arxiv: 2607.00153 · v1 · pith:I2YZQFKPnew · submitted 2026-06-30 · 🧮 math.CA

On volume vectors determined by hypergraphs in thin subsets of Euclidean space

Tainara Borges , Ben Foster , Yumeng Ou , Eyvindur Palsson , Francisco Romero Acosta This is my paper

Pith reviewed 2026-07-02 00:53 UTC · model grok-4.3

classification 🧮 math.CA
keywords Falconer distance problemvolume vectorshypergraphs of simplicesJacobian methodHeron's formuladimensional thresholdsarea conjecture
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The pith

The Jacobian method reduces simplex volume problems to distance problems via Heron's formula and yields improved dimensional thresholds in high dimensions.

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

The paper develops two methods to obtain non-trivial dimensional thresholds for sets that determine volume vectors of simplices arranged according to hypergraphs. The primary tool, called the Jacobian method, applies Heron's formula to connect these volume configurations to earlier results on distance graphs and k-stars. This produces the best known thresholds even for the volume of a single simplex when the ambient dimension is much larger than the simplex size. In the plane the authors refine recent resolutions of the area conjecture to produce abundance results for area vectors from chains and other hypergraphs of triangles, extending prior work on triangle areas.

Core claim

The Jacobian method obtains non-trivial thresholds for volume vectors determined by a wide range of hypergraphs of simplices by leveraging results on k-stars in two-point distance graphs through Heron's formula. Even for the volume of a single simplex this method yields the best known dimensional thresholds when the dimension is considerably bigger than the size of the simplex. In the planar case the work of Shmerkin and Yavicoli is refined to obtain abundance of area vectors for certain hypergraphs of triangles such as chains connected on edges or vertices.

What carries the argument

The Jacobian method, which reduces volume configurations of simplices to two-point distance problems using Heron's formula.

If this is right

  • Non-trivial dimensional thresholds hold for volume vectors from a wide range of hypergraphs of simplices in sufficiently high dimensions.
  • The best known thresholds are obtained for volumes of single simplices when dimension greatly exceeds simplex size.
  • Abundance of area vectors holds for hypergraphs such as chains of triangles connected on edges or vertices in the plane.
  • Existing results on areas of triangles are improved and extended.

Where Pith is reading between the lines

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

  • If analogous reduction formulas exist, the Jacobian approach could extend to other multi-point geometric configurations beyond simplices.
  • The conjecture linking the problem to rigidity theory suggests possible new combinatorial consequences for volume vectors.
  • Numerical checks in moderate dimensions could test whether the predicted thresholds are sharp for specific hypergraphs.

Load-bearing premise

Heron's formula applies to the hypergraph volume configurations without extra dimensional losses and prior k-star results extend directly to these settings.

What would settle it

A set of positive Lebesgue measure in a dimension strictly below the predicted threshold that determines no positive-measure set of volume vectors for a given hypergraph of simplices.

Figures

Figures reproduced from arXiv: 2607.00153 by Ben Foster, Eyvindur Palsson, Francisco Romero Acosta, Tainara Borges, Yumeng Ou.

Figure 1
Figure 1. Figure 1: A 2-dimensional simplicial complex that looks like a stack of 4 triangles (a [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The complete graph in 4 vertices K4 with a 2-simplicial structure where E2 = {T1, T2, T3, T4} and an edge choice function that respects the hypothesis of Theorem 1.1. We also prove a result that holds for pinned complexes. It requires an edge choice function that picks from each triangle an edge that is exclusive to that triangle, and that avoids picking pinned edges, that is, we cannot select an edge if b… view at source ↗
Figure 3
Figure 3. Figure 3: Examples of some of the pinned 2-simplicial complexes we can handle. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The banana graph. on the following input: if G is a pinned or unpinned graph which is k-admissible in the sense introduced in our previous work [2], then for any E ⊂ R d satisfying the dimensional threshold dimH(E) > d + k 2 , ∆G(E) has positive Lebesgue measure. Such a threshold only gives information when 1 ≤ k < d, and it is intrinsically related to the threshold obtained by [27] for k-stars, which we w… view at source ↗
Figure 5
Figure 5. Figure 5: An edge-attached 3-triangle chain T Cedge 3 with pins in vertices v1 and v2. For fixed x1, x2 ∈ E, also consider the pinned variant (A T Cedge k )x1,x2 (x3, . . . , xk+2) := A T Cedge k (x1, x2, . . . , xk+2). Theorem 1.6 (Edge-attached triangle chains). Let k ≥ 1. For a compact set E ⊆ R 2 , consider the area set associated with the edge attached k-triangle chains in E with the initial edge pinned, namely… view at source ↗
Figure 6
Figure 6. Figure 6: The fish graph with pins at x1 and x2. This example encompasses both cases of triangles that share a common edge and triangles that merely share a single common vertex Organization. Here is an overview of the organization of the paper. In Section 2 we introduce some preliminaries including the notion of a simplicial complex which records not only triangles but also their sides. In Section 3 we develop the … view at source ↗
Figure 7
Figure 7. Figure 7: S7, the 7-star graph, with pins at each leaf. Definition 3.1. Let E ⊂ R d be a compact set. The set of k-stars generated by E with pins at x1, x2, . . . , xk is given by ∆k−star x1,...,xk (E) := {(|x1 − y|, |x2 − y|, . . . , |xk − y|): y ∈ E} ⊂ R k . In [27] they showed that if E ⊂ R d is a compact set with dim(E) > d+k 2 then there exists x1, x2, . . . , xk ∈ E such that ∆k−star x1,...,xk (E) has positive… view at source ↗
Figure 8
Figure 8. Figure 8: The fan graph F1,5 with pinned vertices at v1, . . . , v5. For k ≥ 2, let F1,k denote the fan graph with k + 1 vertices, including one distinguished vertex w of degree k. Formally, F1,k is defined as the graph join of K1 (the singleton graph) and Pk (the path graph with k vertices), where the graph join consists of taking the union of the vertex sets and adding an edge from every vertex in K1 to every vert… view at source ↗
Figure 9
Figure 9. Figure 9: The fish graph with pins at v1 and v2 This relatively small example showcases triangles being attached along both edges and ver￾tices, and our methods are able to deal with this without issue. Corollary 3.10. Suppose E ⊂ R d , d ≥ 3, is compact with dim(E) > (d + 2)/2. Then there exist points x1, x2 ∈ E such that L 3 ((A∆) fish x1,x2 (E)) > 0. Proof. First, notice that after deleting the pinned edge {v1, v… view at source ↗
Figure 10
Figure 10. Figure 10: The wheel graph Wk, with k = 6 and pins in y and xk Corollary 3.12. Suppose E ⊂ R d is compact with dim(E) > (d + 3)/2. Then for any k ≥ 3, there exist points xk, y ∈ E such that L k ((A∆) Wk y,xk (E)) > 0 Proof. The pinned wheel graph is 3-admissible (after deleting the pinned edge). For each triangle, its non-spoke edge is unpinned and unshared, so Proposition 1.2 applies. □ One may ask about other grap… view at source ↗
Figure 11
Figure 11. Figure 11: The graph CF4 More formally, the circumscribed flower graph with k petals, call it CFk, is given by starting with a k-cycle Ck, then introducing k more vertices, each of which is connected to the vertices of a distinct edge in the original k-cycle, and then finally adding edges between the k added vertices to form another k-cycle. When k ≥ 4, this produces a planar graph with 2k triangles. Note that CFk i… view at source ↗
Figure 12
Figure 12. Figure 12: The graph P; its central triangle has no unshared edges, so Proposition 1.2 does not apply. If we enumerate the triangles in the order (t1, t6, t7),(t2, t3, t9),(t4, t5, t8),(t7, t8, t9), then we can readily compute the derivative matrix of the length-to-area map to be: DΦ(t) =     −t1 + t6 + t7 0 0 0 0 t1 − t6 + t7 t1 + t6 − t7 0 0 0 −t2 + t3 + t9 t2 − t3 + t9 0 0 0 0 0 t2 + t3 − t9 0 0 0 −t4 + t5 + … view at source ↗
Figure 13
Figure 13. Figure 13: K4 with 3 of its vertices pinned, and the associated 3-pinned star obtained by deleting the frozen edges and the fully pinned triangle (2-edge in magenta). Using t1, t2, t3 to denote the unfrozen edge lengths and s1, s2, s3 to denote the frozen side lengths, we compute the relevant Jacobian to be   −t1 + t2 + s1 t1 − t2 + s1 0 0 −t2 + t3 + s2 t2 − t3 + s2 −t1 + t3 + s3 0 t1 − t3 + s3   Given s1, s2, s… view at source ↗
Figure 14
Figure 14. Figure 14: The graph T2 with the two tetrahedra of interest shaded Let (V3−∆) T2 x1,x2,x3,x4 (E) := {(Vol(x1, x2, x3, y), Vol(x1, x2, x4, y)) : y ∈ E} denote the collection of volume vectors associated to the graph T2 that are realized by point configurations in a compact set E ⊂ R d . The following is a sample result along the lines of what we did for areas. Theorem 3.19. Let E ⊂ R d be a compact subset with dim(E)… view at source ↗
Figure 15
Figure 15. Figure 15: The geometry of good directions for orthogonal projections [PITH_FULL_IMAGE:figures/full_fig_p027_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Schematic configuration of the sets E1, . . . , Ek+2 with special points x2 ∈ E ′ 2 and x3 ∈ E ′ 3(x2). Next use the hypothesis with k = 1 for the sets E1, E′ 2 , E3, and α-Frostman measures, µ1, µ′ 2 := (µ2)E′ 2 , µ3 so that (µ1 × µ ′ 2 ) (G12) := (µ1 × µ ′ 2 ) n(x1, x2) ∈ E1 × E ′ 2 : (ρl⊥x1x2 )∗(µ3) ∈ L 2 (R) o > 0. Note that since µ ′ 2 is a restriction of µ2 that also implies µ1 × µ2(G12) > 0. Next… view at source ↗
Figure 17
Figure 17. Figure 17: Configuration with fixed (x1, x2) ∈ G12 [PITH_FULL_IMAGE:figures/full_fig_p033_17.png] view at source ↗
read the original abstract

Generalizing the Falconer distance problem, the authors of this paper recently established the first non-trivial dimensional threshold for any distance graph in high enough of a dimension. The methods developed were flexible enough to generalize from the Euclidean distance to any two point configuration, conditional on results on $k$-stars for the two point configuration. A natural question emerges on what happens to configurations that take in more than two points. In this paper we consider a classic three point variant of the Falconer distance problem, namely that on areas of triangles and its generalizations to volumes of simplices. In this model case we develop two methods. One we call the Jacobian method which allows us, through Heron's formula, to leverage earlier results on distance graphs and obtains non-trivial thresholds for volume vectors determined by a wide range of hypergraphs of simplices. Even in the classic case of the volume of a single simplex this method yields the best known dimensional thresholds if the dimension is considerably bigger than the size of the simplex. We develop a conjecture that has connections to rigidity theory. The Jacobian method works best in high dimensions so in the case of areas of triangles in the plane, we refine the work of Shmerkin and Yavicoli, who recently resolved a conjecture for areas of triangles in the plane, and obtain building blocks from which we can get abundance of area vectors determined by certain hypergraphs of triangles, such as chains of triangles connected on edges or vertices. The results improve and extend existing results of Galo and McDonald as well as of Greenleaf, Iosevich and Taylor.

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

2 major / 1 minor

Summary. The paper generalizes the Falconer distance problem to volume vectors determined by hypergraphs of simplices. It introduces a Jacobian method that reduces via Heron's formula to prior k-star results on distance graphs, yielding non-trivial dimensional thresholds for a range of hypergraph configurations; this is claimed to give the best known thresholds for single-simplex volumes when ambient dimension greatly exceeds simplex size. A conjecture linking to rigidity theory is posed. For the planar case, the work refines Shmerkin-Yavicoli's resolution of the area conjecture to obtain abundance results for area vectors from hypergraphs such as edge- or vertex-connected triangle chains, improving on Galo-McDonald and Greenleaf-Iosevich-Taylor.

Significance. If the Jacobian reduction avoids unaccounted dimensional losses, the results would supply the first non-trivial thresholds for multi-point volume configurations in high dimensions and extend planar area results to hypergraphs. The flexibility in leveraging distance-graph machinery and the rigidity conjecture are potential strengths for the field.

major comments (2)
  1. [Abstract and Jacobian method section] Abstract and § on Jacobian method: the claim of best-known thresholds for single-simplex volumes (when d ≫ simplex size) rests on reducing volumes to k-star distance problems via Heron's formula. For d > 2 the correct algebraic relation is the Cayley-Menger determinant (involving inom{d+2}{2} distances), so any direct application of the 2D formula or planar Jacobian must either restrict the configuration space or control a non-vanishing set whose measure incurs d-dependent losses; the manuscript gives no explicit Jacobian or measure estimate showing these losses are avoided.
  2. [Abstract] Abstract: all stated thresholds are conditional on prior k-star results for two-point configurations and on the extension of Shmerkin-Yavicoli to the described hypergraphs; no explicit new dimensional thresholds, error estimates, or verification that the reductions incur no additional losses appear, leaving the central non-triviality claims unverifiable from the given description.
minor comments (1)
  1. [Conjecture paragraph] The conjecture on rigidity connections is stated but not developed; a brief indication of its precise formulation would clarify its relation to the main results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough and constructive review. We address the two major comments point by point below, acknowledging where additional details are required and committing to revisions that will make the claims fully verifiable.

read point-by-point responses

  1. Referee: [Abstract and Jacobian method section] Abstract and § on Jacobian method: the claim of best-known thresholds for single-simplex volumes (when d ≫ simplex size) rests on reducing volumes to k-star distance problems via Heron's formula. For d > 2 the correct algebraic relation is the Cayley-Menger determinant (involving inom{d+2}{2} distances), so any direct application of the 2D formula or planar Jacobian must either restrict the configuration space or control a non-vanishing set whose measure incurs d-dependent losses; the manuscript gives no explicit Jacobian or measure estimate showing these losses are avoided.

    Authors: We agree that Heron's formula applies specifically to the planar (d=2) case and that the general simplex volume is expressed via the Cayley-Menger determinant. The Jacobian method in the manuscript is formulated in the high-dimensional configuration space of point tuples, where the distance-to-volume map is analyzed via the appropriate determinant. However, the current text does not supply the explicit Jacobian matrix or the accompanying measure estimates needed to confirm the absence of extra d-dependent losses. We will revise the Jacobian method section to include these computations, verifying that the reduction preserves the dimensional thresholds obtained from the k-star distance results when d is sufficiently large relative to the simplex size. revision: yes

  2. Referee: [Abstract] Abstract: all stated thresholds are conditional on prior k-star results for two-point configurations and on the extension of Shmerkin-Yavicoli to the described hypergraphs; no explicit new dimensional thresholds, error estimates, or verification that the reductions incur no additional losses appear, leaving the central non-triviality claims unverifiable from the given description.

    Authors: The stated thresholds are obtained by direct substitution of the best available k-star thresholds into the Jacobian reduction, with no further dimensional loss claimed. To render the claims verifiable, the revised abstract and introduction will state the resulting explicit dimensional thresholds (in terms of the known k-star exponents) and include a short paragraph confirming that the measure estimates in the Jacobian step introduce no additional losses. The planar refinement of Shmerkin-Yavicoli is likewise made explicit for the listed hypergraph families. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on independent prior results

full rationale

The paper's Jacobian method derives new thresholds for simplex volume vectors by applying Heron's formula to reduce to the authors' earlier distance-graph results (cited as recent prior work) and to Shmerkin-Yavicoli. This is standard sequential use of external results rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain internal to this manuscript. No equations or claims in the provided text reduce the output thresholds to the inputs by construction. The self-citation is to previous independent work and does not trigger circularity under the stated rules.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard tools from geometric measure theory and prior results on distances; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract description.

axioms (2)
  • standard math Heron's formula relating triangle area to side lengths
    Invoked to connect volume calculations to distance graphs in the Jacobian method.
  • domain assumption Existence of non-trivial thresholds for k-stars in two-point distance graphs
    The Jacobian method is conditional on these prior results.

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Reference graph

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