pith. sign in

arxiv: 2605.18558 · v1 · pith:4X65EJEUnew · submitted 2026-05-18 · 🧮 math.GT

Normalized volume spectra of right-angled hyperbolic polyhedra

A. Egorov , A. Vesnin This is my paper

Pith reviewed 2026-05-20 08:15 UTC · model grok-4.3

classification 🧮 math.GT
keywords right-angled hyperbolic polyhedranormalized volumevolume spectrumideal polyhedracompact polyhedrahyperbolic geometrydiscrete spectrumdense spectrum
0
0 comments X

The pith

The normalized volumes of right-angled hyperbolic polyhedra occupy bounded intervals that are discrete at the bottom and dense near the top.

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

The paper defines the normalized volume of a hyperbolic polyhedron as its total volume divided by its number of vertices. It then collects these values into spectra for two families: all compact right-angled hyperbolic polyhedra and all ideal right-angled hyperbolic polyhedra. For the ideal family the spectrum fills the closed interval from one-sixth to one-half the volume of the regular ideal octahedron, with the lower part discrete and the upper part dense. For the compact family the spectrum fills an interval from roughly 5/192 of the octahedron volume up to five-eighths the volume of the regular ideal tetrahedron, again discrete at the bottom and dense near the upper end. These statements give precise control over which average volumes per vertex are achievable by right-angled hyperbolic polyhedra of either type.

Core claim

The spectrum Ω(R_ideal) belongs to the interval [1/6 v_oct, 1/2 v_oct] and both bounds are sharp; the spectrum is discrete in [1/6 v_oct, 1/4 v_oct) and everywhere dense in [1/4 v_oct, 1/2 v_oct]. The spectrum Ω(R_comp) belongs to [5/192 v_oct, 5/8 v_tet] with the upper bound sharp; it is discrete on [5/192 v_oct, 1/32 v_oct) and dense on [5/16 v_tet, 5/8 v_tet].

What carries the argument

The normalized volume ω(P) = vol(P)/ver(P), which rescales each polyhedron's volume by its vertex count so that volumes of polyhedra with different numbers of vertices become comparable.

If this is right

  • Any ideal right-angled hyperbolic polyhedron has normalized volume at most half the volume of the regular ideal octahedron.
  • Below one-quarter of the octahedron volume the possible normalized volumes for ideal polyhedra occur only at isolated points.
  • Normalized volumes of ideal polyhedra can be made arbitrarily close to the upper bound of one-half the octahedron volume.
  • Compact right-angled hyperbolic polyhedra cannot exceed five-eighths the volume of the regular ideal tetrahedron in normalized volume.
  • In the upper part of their range the normalized volumes of compact polyhedra become dense.

Where Pith is reading between the lines

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

  • The discreteness results suggest that only finitely many combinatorial types realize each small normalized volume, which could be checked by enumerating low-vertex right-angled polyhedra.
  • The density statements imply that one can approximate any volume in the dense interval by a right-angled polyhedron, potentially useful for constructing manifolds with prescribed volume properties.
  • The two families (ideal and compact) have overlapping but distinct normalized-volume ranges, hinting at a possible transition when vertices move from ideal to finite positions.

Load-bearing premise

Sequences of right-angled hyperbolic polyhedra exist that realize the stated upper bounds and fill the dense portions of the intervals while keeping all dihedral angles right and the geometry hyperbolic.

What would settle it

A single right-angled ideal hyperbolic polyhedron whose normalized volume lies strictly outside [1/6 v_oct, 1/2 v_oct], or a right-angled compact hyperbolic polyhedron whose normalized volume lies strictly outside [5/192 v_oct, 5/8 v_tet].

Figures

Figures reproduced from arXiv: 2605.18558 by A. Egorov, A. Vesnin.

Figure 1
Figure 1. Figure 1: Antiprisms A(3) and A(4), and the twisted antiprism A(4)∗ . Following [11], define an operation called the edge-twist. The operation is as follows. Let P be an abstract polyhedron that has, in some face, four distinct 4-valent vertices forming two pairs of adjacent vertices joined by edges e1 and e2. Apply to the face the transformation consisting in deleting the edges e1 and e2, creating a new vertex v, a… view at source ↗
Figure 2
Figure 2. Figure 2: Edge-twist of the edges e1 and e2. by P ∗ . We will say that P ∗ is obtained from P by the edge-twist. In [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Initial part of the growth graph of polyhedra from A(4). vol(A(4)∗ ) = 2vol(A(3)). The volumes of the polyhedra A(3) and A(4) are computed by formula (2). An example of a decrease in the normalized volume is the pair of polyhedra marked in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The polyhedra P5 and P8 = P ∗ 5 . Upper and lower estimates of the volumes of ideal right-angled polyhedra in terms of the number of vertices were obtained by Atkinson in [7]. Theorem 2.3. [7, Theorem 2.2] If an ideal right-angled hyperbolic polyhedron P has ver(P) vertices, then (4) voct 4 (ver(P) − 2) ≤ vol(P) ≤ voct 2 (ver(P) − 4) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The dodecahedron L(5), the polyhedron L(6), and the polyhedron L(6)+. As shown in [32], the volume of the polyhedron L(n), n ≥ 5, is expressed by the following formula: (6) vol(L(n)) = n 2 h 2Λ(θn) + Λ θn + π n  + Λ θn − π n  − Λ  2θn − π 2 i , where θn = π 2 − arccos  1 2 cos(π/n)  . In particular, vol(L(5)) = 4, 306210 and vol(L(6)) = 6, 023046. Geometric and arithmetic properties of the polyhedr… view at source ↗
Figure 6
Figure 6. Figure 6: Edge surgery along the edge e. by P + = P ∪ {e} [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Initial part of the growth graph of polyhedra from L(6). e1 e2 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The polyhedra Q and Q+. Below we will use L¨obell polyhedra L(n) to construct new compact right-angled polyhedra. Example 3.2. For an integer k ≥ 1, denote by Lk(n) the polyhedron with 2n(k + 1) vertices constructed from k copies of the polyhedron L(n) by their successive gluing along n-gonal faces. In particular, L1(n) = L(n). As an example, [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The lateral surface of the polyhedron L3(6). In particular, for k = 1 we obtain limn→∞ ω(L(n)) = 5 16 vtet, and as k → ∞ we have lim k,n→∞ ω(Lk(n)) = 5 8 vtet. Volume estimates for compact right-angled polyhedra were obtained by Atkinson [7]. Theorem 3.3. [7, Theorem 2.3]. If a compact right-angled hyperbolic polyhedron P has ver(P) vertices, then (8) voct 32 (ver(P) − 8) ≤ vol(P) < 5vtet 8 (ver(P) − 10). … view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of spectra of normalized volumes. The corresponding intervals are shown in [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
read the original abstract

Let a three-dimensional hyperbolic polyhedron $\mathcal P$ have finite volume $\mathrm{vol}(\mathcal P)$ and a finite number of vertices $\mathrm{ver}(\mathcal P)$. We call its normalized volume the quantity $\omega(\mathcal P) = \mathrm{vol}(\mathcal P)/ \mathrm{ver}(\mathcal P)$. If $\mathcal{R}$ is some set of hyperbolic polyhedra, then we assign to it the set of normalized volumes $\Omega (\mathcal R) = \{ \omega(\mathcal P) \mid \mathcal P \in \mathcal R \}$, which we call the spectrum of normalized volumes of the set $\mathcal R$. In the paper we consider the set $\mathcal R_{comp}$ of compact right-angled hyperbolic polyhedra and the set $\mathcal R_{ideal}$ of ideal right-angled hyperbolic polyhedra. We prove that the spectrum $\Omega (\mathcal R_{ideal})$ belongs to the interval $\left[\frac{1}{6} v_{\mathrm{oct}}, \frac{1}{2} v_{\mathrm{oct}} \right]$ and both bounds are sharp. Moreover, the spectrum is discrete in $\left[ \frac{1}{6} v_{\mathrm{oct}}, \frac{1}{4} v_{\mathrm{oct}} \right)$ and everywhere dense in $\left[ \frac{1}{4} v_{\mathrm{oct}}, \frac{1}{2} v_{\mathrm{oct}} \right]$, where $v_{\mathrm{oct}}$ is the volume of the regular ideal hyperbolic octahedron. We also establish that the spectrum $\Omega (\mathcal R_{comp})$ belongs to the interval $\left[ \frac{5}{192} v_{\mathrm{oct}}, \frac{5}{8} v_{\mathrm{tet}} \right]$ and the upper bound is sharp. Moreover, on the interval $\left[ \frac{5}{192} v_{\mathrm{oct}}, \frac{1}{32} v_{\mathrm{oct}} \right)$ the spectrum is discrete, while on the interval $\left[ \frac{5}{16} v_{\mathrm{tet}}, \frac{5}{8} v_{\mathrm{tet}} \right]$ it is everywhere dense, where $v_{\mathrm{tet}}$ is the volume of the regular ideal hyperbolic tetrahedron.

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 / 2 minor

Summary. The paper defines the normalized volume ω(P) = vol(P)/ver(P) for finite-volume hyperbolic polyhedra P with finitely many vertices. It studies the spectra Ω(R_ideal) and Ω(R_comp) consisting of all such normalized volumes for the classes of ideal right-angled hyperbolic polyhedra and compact right-angled hyperbolic polyhedra, respectively. The main theorems assert that Ω(R_ideal) lies in [1/6 v_oct, 1/2 v_oct] with both endpoints achieved, is discrete on [1/6 v_oct, 1/4 v_oct), and is dense on [1/4 v_oct, 1/2 v_oct]; likewise Ω(R_comp) lies in [5/192 v_oct, 5/8 v_tet] with the upper endpoint achieved, is discrete on [5/192 v_oct, 1/32 v_oct), and is dense on [5/16 v_tet, 5/8 v_tet].

Significance. If the stated bounds, sharpness statements, and discreteness/density dichotomy hold, the work supplies a precise structural description of the possible normalized volumes attainable by right-angled hyperbolic polyhedra. Such polyhedra serve as fundamental building blocks for hyperbolic 3-manifolds and orbifolds; a complete understanding of their normalized-volume spectra would therefore constrain the geometry of finite-volume hyperbolic structures and could inform questions about volume minimization or rigidity in the presence of right angles. The explicit constants involving the regular ideal octahedron and tetrahedron volumes, together with the transition point separating discrete and dense regimes, constitute concrete, falsifiable predictions.

major comments (2)
  1. [section on upper bound for ideal spectrum] Proof of sharpness of the upper bound 1/2 v_oct for Ω(R_ideal) (abstract and the section containing the construction of the maximizing polyhedron): the argument must explicitly confirm that the limiting or glued object remains right-angled, i.e., that every dihedral angle stays exactly π/2 under the deformation or gluing parameters used to reach the bound. Without an independent verification that the angle condition is preserved (rather than merely controlling volume or cusp shape), the constructed object may exit R_ideal.
  2. [section on density in the upper interval for ideal polyhedra] Density statement for Ω(R_ideal) on [1/4 v_oct, 1/2 v_oct] (the section constructing the dense sequence P_n with ver(P_n)→∞): the limiting process must be shown to keep all dihedral angles fixed at π/2. If the parameters are chosen solely to adjust volume while the right-angled condition is only verified for base cases, the limit objects may fail to lie in R_ideal, undermining the density claim.
minor comments (2)
  1. [abstract] The constants v_oct and v_tet should be defined at their first appearance in the abstract and introduction, together with a brief reminder of their numerical values or explicit formulas.
  2. [introduction] Notation for the sets R_comp and R_ideal is introduced only in the abstract; a short paragraph in the introduction repeating the definitions would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on the preservation of the right-angled condition. We address each major comment below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: Proof of sharpness of the upper bound 1/2 v_oct for Ω(R_ideal) (abstract and the section containing the construction of the maximizing polyhedron): the argument must explicitly confirm that the limiting or glued object remains right-angled, i.e., that every dihedral angle stays exactly π/2 under the deformation or gluing parameters used to reach the bound. Without an independent verification that the angle condition is preserved (rather than merely controlling volume or cusp shape), the constructed object may exit R_ideal.

    Authors: We appreciate this observation. Our construction for the upper bound proceeds by gluing copies of the regular ideal octahedron (which has all dihedral angles π/2) along entire faces. Because the gluings identify faces with matching right angles and introduce no new edges with non-right angles, the resulting polyhedron remains in R_ideal at each finite stage; the limiting normalized volume is therefore attained within the class. We will add a dedicated paragraph in the relevant section that explicitly verifies dihedral-angle preservation under the gluing parameters, independent of volume control. revision: yes

  2. Referee: Density statement for Ω(R_ideal) on [1/4 v_oct, 1/2 v_oct] (the section constructing the dense sequence P_n with ver(P_n)→∞): the limiting process must be shown to keep all dihedral angles fixed at π/2. If the parameters are chosen solely to adjust volume while the right-angled condition is only verified for base cases, the limit objects may fail to lie in R_ideal, undermining the density claim.

    Authors: We agree that an explicit check is needed. The sequence P_n is obtained by successive combinatorial gluings of right-angled polyhedra (starting from the octahedron and smaller right-angled pieces) along faces; each gluing preserves dihedral angles of π/2 by the local geometry of the faces. The parameters control only the number of vertices and total volume while the angle condition is maintained inductively. We will revise the section to include a short inductive argument confirming that every P_n lies in R_ideal, so the accumulation points of their normalized volumes lie in the claimed dense interval. revision: yes

Circularity Check

0 steps flagged

No circularity; results derived from explicit geometric constructions and volume calculations in hyperbolic 3-space.

full rationale

The paper defines normalized volume ω(P) = vol(P)/ver(P) directly from hyperbolic volume and vertex count, then proves interval membership, sharpness, discreteness, and density for Ω(R_ideal) and Ω(R_comp) via constructions of right-angled polyhedra, gluings, and limiting processes that preserve dihedral angles π/2. These steps rely on standard hyperbolic geometry (e.g., ideal octahedron and tetrahedron volumes) and explicit families rather than any fitted parameter renamed as prediction, self-definitional loop, or load-bearing self-citation that reduces the central claims to inputs. The derivation chain is self-contained against external hyperbolic volume formulas and does not invoke uniqueness theorems or ansatzes from the authors' prior work in a circular manner.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Claims rely on standard properties of hyperbolic 3-space, existence of right-angled polyhedra with prescribed combinatorics, and volume formulas for ideal and compact cases; no free parameters or new entities are introduced.

axioms (1)
  • standard math Standard axioms and volume properties of hyperbolic 3-geometry
    Invoked throughout for defining volumes, ideal points, and right angles in hyperbolic space.

pith-pipeline@v0.9.0 · 5976 in / 1378 out tokens · 44172 ms · 2026-05-20T08:15:47.128912+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    Adams, A

    C. Adams, A. Calderon, N. Mayer,Generalized bipyramids and hyperbolic volumes of alternating k–uniform tiling links, Topology and its Applications271(2020), 107045. 2

    work page 2020

  2. [2]

    Adams, F

    C. Adams, F. Gomez-Raz, J. Kang, L. Krause, G. Li, C. Marple, Z. Tan,Hyperbolic handlebody complements in 3-manifolds, Topology and its Applications377(2026), 109642. 19

    work page 2026

  3. [3]

    D. V. Alekseevskij, E. B. Vinberg, A. S. Solodovnikov,Geometry of spaces of constant curvature, in: Geometry II, Encyclopedia Math. Sci.29, Springer, Berlin (1993), 1–138. 3

    work page 1993

  4. [4]

    S. A. Alexandrov, N. V. Bogachev, A. Yu. Vesnin, A. A. Egorov,On volumes of hyperbolic right-angled polyhedra, Sb. Math.214:2(2023), 148–165. 7, 11

    work page 2023

  5. [5]

    E. M. Andreev,On convex polyhedra of finite volume in Lobachevskii space, Math. USSR-Sb.12:2(1970), 255–259. 3

    work page 1970

  6. [6]

    E. M. Andreev,On convex polyhedra in Lobachevskii spaces, Math. USSR-Sb.10:3(1970), 413–440. 3

    work page 1970

  7. [7]

    Atkinson,Volume estimates for equiangular hyperbolic Coxeter polyhedra, Algebraic & Geometric Topology 9(2009), 1225–1254

    C. Atkinson,Volume estimates for equiangular hyperbolic Coxeter polyhedra, Algebraic & Geometric Topology 9(2009), 1225–1254. 2, 3, 6, 11

    work page 2009

  8. [8]

    Belletti,The maximum volume of hyperbolic polyhedra, Trans

    G. Belletti,The maximum volume of hyperbolic polyhedra, Trans. Amer. Math. Soc.374(2)(2021), 1125–1153. 1

    work page 2021

  9. [9]

    Bogachev, S

    N. Bogachev, S. Douba,Geometric and arithmetic properties of L¨ obell polyhedra, Algebr. Geom. Topol.25:4 (2025), 2281–2295. 8

    work page 2025

  10. [10]

    A. A. Borisenko, A. Yu. Vesnin, N. M. Ivochkina,On the 100th anniversary of the birth of Aleksei Vasil’evich Pogorelov, Russian Math. Surveys74:6(2019), 1135–1157. 8

    work page 2019

  11. [11]

    Brinkmann, S

    G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas, P. Wollan,Generation of simple quad- rangulations of the sphere, Discrete Mathematics305(2005), 33–54. 4, 5

    work page 2005

  12. [12]

    S. D. Burton,The spectra of volume and determinant densities of links, Topology and its Applications211 (2016), 38–55. 2 20 A. EGOROV, A. VESNIN

    work page 2016

  13. [13]

    Buser, A

    P. Buser, A. Mednykh, A. Vesnin,Lambert cubes and L¨ obell polyhedron revised, Advances in Geometry12 (2012), 525–548. 8

    work page 2012

  14. [14]

    Champanerkar, I

    A. Champanerkar, I. Kofman, J. S. Purcell,Geometrically and diagrammatically maximal knots, Journal of the London Mathematical Society94:3(2016), 883–908. 2

    work page 2016

  15. [15]

    Costantino, F

    F. Costantino, F. Gueritaud, R. van der Veen,On the volume conjecture for polyhedra, Geom. Dedicata179 (2015), 385–409. 1

    work page 2015

  16. [16]

    Egorov, A

    A. Egorov, A. Vesnin,On correlation of hyperbolic volumes of fullerenes with their properties, Comput. Math. Biophys.8(2020), 150–167. 8

    work page 2020

  17. [17]

    Egorov, A

    A. Egorov, A. Vesnin,Volume estimates for right-angled hyperbolic polyhedra, Rend. Istit. Mat. Univ. Trieste 52(2020), 565–576. 7, 11

    work page 2020

  18. [18]

    Futer, F

    D. Futer, F. Gueritaud,From angled triangulations to hyperbolic structures, Interactions between hyperbolic geometry, quantum topology and number theory, Contemp. Math.541, Amer. Math. Soc., Providence, RI, 2011, 159–182. 13

    work page 2011

  19. [19]

    Haagerup, H

    U. Haagerup, H. Munkholm,Simplices of maximal volume in hyperbolic n-space, Acta Math.147(1981), 1–11. 1

    work page 1981

  20. [20]

    Inoue,Organizing volumes of right-angled hyperbolic polyhedra, Algebraic & Geometric Topology8(2008), 1523–1565

    T. Inoue,Organizing volumes of right-angled hyperbolic polyhedra, Algebraic & Geometric Topology8(2008), 1523–1565. 7, 8, 9

    work page 2008

  21. [21]

    Inoue,Exploring the list of smallest right-angled hyperbolic polyhedra, Experimental Math.31:1(2022), 165–183

    T. Inoue,Exploring the list of smallest right-angled hyperbolic polyhedra, Experimental Math.31:1(2022), 165–183. 2, 9

    work page 2022

  22. [22]

    Kwon,Fully augmented links in the thickened torus, Algebraic & Geometric Topology25:3(2025), 1411–1432

    A. Kwon,Fully augmented links in the thickened torus, Algebraic & Geometric Topology25:3(2025), 1411–1432. 3

    work page 2025

  23. [23]

    A. Kwon, Y. H. Tham,On the volume density spectrum of fully augmented links, Journal of Knot Theory and Its Ramifications34:05(2025), 2550011. 2, 17

    work page 2025

  24. [24]

    Lackenby,The volume of hyperbolic alternating link complements

    M. Lackenby,The volume of hyperbolic alternating link complements. With an appendix by I. Agol and D. Thurston, Proc. London Math. Soc.88(2004), 204–224. 3, 4, 17

    work page 2004

  25. [25]

    Milnor,Hyperbolic geometry: the first 150 years, Bull

    J. Milnor,Hyperbolic geometry: the first 150 years, Bull. Amer. Math. Soc.6(1982), 9–24. 1, 5

    work page 1982

  26. [26]

    A. V. Pogorelov,A regular partition of Lobachevskian space, Math. Notes1:1(1967), 3–5. 8

    work page 1967

  27. [27]

    V. V. Prasolov.Elements of combinatorial and differential topology. AMS, 2022. 3

    work page 2022

  28. [28]

    J. S. Purcell,An introduction to fully augmented links, Interactions between hyperbolic geometry, quantum topology and number theory, Contemp. Math.541, Amer. Math. Soc., Providence, RI, 2011, 205–220. 3, 4, 17

    work page 2011

  29. [29]

    R. K. W. Roeder, J. H. Hubbard, W. D. Dunbar,Andreev’s theorem on hyperbolic polyhedra, Ann. Inst. Fourier Grenoble57(2007) 825–882. 3

    work page 2007

  30. [30]

    W. P. Thurston,The Geometry and Topology of Three-Manifolds, Collected Works of William P. Thurston with Commentary, Vol. IV, With a preface by S. P. Kerckhoff, Edited by B. Farb, D. Gabai and S. P. Kerckhoff, American Mathematical Society, Providence, RI, 2022. 5

    work page 2022

  31. [31]

    A. Yu. Vesnin,Three-dimensional hyperbolic manifolds of L¨ obell type, Siberian Math. J.28:5(1987), 731–734. 8

    work page 1987

  32. [32]

    A. Yu. Vesnin,Volumes of hyperbolic L¨ obell 3-manifolds, Math. Notes64:1(1998), 15–19. 8

    work page 1998

  33. [33]

    Vesnin,Volumes and Normalized Volumes of Right-Angled Hyperbolic Polyhedra, Atti Semin

    A. Vesnin,Volumes and Normalized Volumes of Right-Angled Hyperbolic Polyhedra, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia57(2010), 159–169. 2

    work page 2010

  34. [34]

    A. Yu. Vesnin,Right-angled polyhedra and hyperbolic 3-manifolds, Russian Math. Surveys72:2(2017), 335–374. 1

    work page 2017

  35. [35]

    Vesnin, A

    A. Vesnin, A. Egorov,Ideal right-angled polyhedra in Lobachevsky space, Chebyshevskii Sbornik21:2(2020), 65–83. 2, 6, 7

    work page 2020

  36. [36]

    A. Yu. Vesnin, A. A. Egorov,Upper bounds for volumes of generalized hyperbolic polyhedra and hyperbolic links, Siberian Math. J.65:3(2024), 534–551. 3, 4, 17

    work page 2024

  37. [37]

    A. Yu. Vesnin, A. A. Egorov,The right-angled Coxeter group of minimal covolume in three-dimensional hyper- bolic space, Siberian Math. J.66:6(2025), 1374–1389. 19

    work page 2025

  38. [38]

    A. Yu. Vesnin, D. Repovˇ s,Two-sided bounds for the volume of right-angled hyperbolic polyhedra, Math. Notes 89:1(2011), 31–36. 10 Sobolev Institute of Mathematics, Novosibirsk, Russia Email address:a.egorov2@g.nsu.ru Sobolev Institute of Mathematics, Novosibirsk, Russia Email address:vesnin@math.nsc.ru

    work page 2011