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APPEARED IN BULLETIN OF THE

arXiv:math/9210218v1 [math.MG] 1 Oct 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 27, Number 2, October 1992, Pages 246-251A CHARACTERIZATION OF CONVEX HYPERBOLICPOLYHEDRA AND OF CONVEX POLYHEDRA INSCRIBED INTHE SPHERECRAIG D. HODGSON, IGOR RIVIN, AND WARREN D. SMITHAbstract. We describe a characterization of convex polyhedra in H3 in termsof their dihedral angles, developed by Rivin.

We also describe some geometricand combinatorial consequences of that theory. One of these consequences is acombinatorial characterization of convex polyhedra in E3 all of whose verticeslie on the unit sphere.That resolves a problem posed by Jakob Steiner in1832.In 1832, Jakob Steiner in his book [23] asked the following question:In which cases does a convex polyhedron have a (combinatorial)equivalent which is inscribed in, or circumscribed about, a sphere?This was the 77th of a list of 85 open problems posed by Steiner, of which onlynumbers 70, 76, and 77 were still open as of last year.

Apparently Ren´e Descarteswas also interested in the problem (see [12]).Several authors found families of noninscribable polyhedral types, beginning withSteinitz in 1927 (cf. [14]); all of these families later were subsumed by a theoremof Dillencourt [11].

In their 1991 book [9, problem B18], Croft, Falconer, and Guyhad the following to say:It would of course be nice to characterize the polyhedra of inscrib-able type, but as this may be over-optimistic, good necessary, orsufficient, conditions would be of interest.Here we announce a full answer to Steiner’s question, in the sense that we producea characterization of inscribable (or circumscribable) polyhedra that has a numberof pleasant properties—it can be checked in polynomial time and it yields a numberof combinatorial corollaries. First we note the following well-known characterizationof convex polyhedra proved by Steinitz (cf.

[14]).Theorem of Steinitz. A graph is the one-skeleton of a convex polyhedron in E3if and only if it is a 3-connected planar graph.Note.

A graph G is k-connected if the complement of any k −1 edges in G isconnected.Received by the editors August 30, 1991.1991 Mathematics Subject Classification. Primary 52A55, 53C45, 51M20; Secondary 51M10,05C10, 53C50.c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2C. D. HODGSON, IGOR RIVIN, AND W. D. SMITHWe will call graphs satisfying the criteria of Steinitz’ theorem polyhedral graphs.The answer to Steiner’s question stems from the following characterization ofideal convex polyhedra in hyperbolic 3-space H3.

(See [25, 7] for the basics ofhyperbolic geometry. )Theorem 1.

Let P be a polyhedral graph with weights w(e) assigned to the edges.Let P ∗be the planar dual (or Poincar´e dual) of P, where the edge e∗dual to e isassigned the dual weight w∗(e∗) = π −w(e). Then P can be realized as a convexpolyhedron in H3 with all vertices on the sphere at infinity and with dihedral anglew(e) at every edge e if and only if the following conditions hold:(1) 0 < w∗(e∗) < π for all edges e.(2) The sum of dual weights of edges e∗1, e∗2, .

. .

, e∗k bounding a face in P ∗isequal to 2π. (3) The sum of dual weights of edges e∗1, e∗2, .

. .

, e∗k forming a circuit that doesnot bound a face in P ∗is strictly greater than 2π.Theorem 2. A realization guaranteed by Theorem 1 is unique up to isometries ofH3.Theorem 1 is proved by Rivin in [22].

It uses the methods of Aleksandrov [4]and also results and methods developed by Rivin in [18, 17] and subsequent work.A brief introduction to this theory is given in §1. A more complete treatment isgiven in [21].Notes.

Theorems 1 and 5 were recently extended by Rivin to general hyperbolicpolyhedra of finite volume (that is, those with some finite and some ideal vertices).A characterization of ideal polyhedra with dihedral angles not greater than π/2 wasgiven by Andreev [6]; Andreev’s result is an easy consequence of Theorem 1.The so-called projective model (or Klein model) of hyperbolic 3-space is a rep-resentation of H3 as the interior of the unit ball B3 in the ordinary Euclidean3-space E3. The model has the property of being geodesic—hyperbolic lines andplanes are represented by Euclidean lines and planes, respectively.

Convexity isalso preserved—a convex body in H3 is represented by a convex body in B3. Thus,hyperbolic convex polyhedra with all vertices on the sphere at infinity correspondprecisely to convex Euclidean polyhedra inscribed in the sphere S2 = ∂B3.

There-fore, a polyhedron is of inscribable type exactly when it admits an edge-weightingthat satisfies the condition in Theorem 1.Furthermore (see [14]), a polyhedron is inscribable if and only if its planar dualis circumscribable, so we can sum up the characterization as follows.Characterization R∗. A polyhedron P is of circumscribable type if and only ifthere exists a weighting w of its edges, such that:(1) The weight of any edge satisfies 0 < w(e) < 1/2.

(2) The total weight of a boundary of a face of P is equal to 1. (3) The total weight of any circuit not bounding a face is strictly greater than1.Characterization R. A polyhedron P is of inscribable type if and only if its planardual satisfies the conditions (1)–(3) of Characterization R∗.The following theorem was proved by Smith:

CONVEX HYPERBOLIC AND CONVEX POLYHEDRA IN THE SPHERE3Theorem 3. Given a polyhedral graph P, we can decide whether it admits a weight-ing satisfying Characterization R∗in time polynomial in the number of verticesN.

More exactly : on an integer Random Access Memory (RAM ) Machine (see[1]) with precision bounded by O(log N) bits, the running time may be bounded byO(N 5.38) operations.Skeleton of Proof. Finding the desired weighting is a linear program with the num-ber of constraints exponential in N and the methods of [13] and [26] can be used toproduce the algorithm of Theorem 3.

The algorithm exploits the observation thatgiven a graph with prescribed weights on the edges, it is possible to determine inpolynomal time whether the weights satisfy conditions (1)–(3) of CharacterizationR∗. Given that, a variant of the Ellipsoid Method is seen to yield the desired algo-rithm.

Results of [26] allow us to improve the asymptotic behavior of the algorithmsomewhat; the funny looking exponent 5.38 stems from the best known complexityresult for matrix inversion.Note (added in proof). Rivin [20] recently found a much smaller (linear in N) linearprogram, and hence a simpler algorithm.Hence, the two realizability questions above may also be answered in polynomialtime.

For some special classes of graphs, it is particularly easy to decide inscriba-bility. We mention the following theorem of M. Dillencourt:Any polyhedron whose graph is 4-connected, is inscribable.

Also,these graphs are circumscribable. More graph-theoretic results canbe found in [10].1.

Characterization of hyperbolic polyhedraThe work of Aleksandrov [3, 4] gives a complete characterization of compactconvex polyhedra in hyperbolic 3-space in terms of the intrinsic hyperbolic metricon the boundary. Note: Aleksandrov’s work has now been extended by Rivin [19]to ideal convex polyhedra.Theorem 5 gives an analogous characterization of convex hyperbolic polyhedrain terms of their dihedral angles.

This also generalizes the work of Andreev [5]. Asimple derivation of Andreev’s results from Theorem 5 is given by Hodgson in [15].1.1.

Compact polyhedra. The material from this section is developed in [18].See [21] for a more detailed exposition.

Consider the Gauss Map G of a compactconvex polyhedron P in Euclidean three-dimensional space E3. The map G is aset-valued function from P to the unit sphere S2, which assigns to each point p theset of outward unit normals to support planes to P at p. Thus, the whole of a face fof P is mapped under G to a single point—the outward unit normal to f. An edgee of P is mapped to a geodesic segment G(e) on S2, whose length is easily seen tobe the exterior dihedral angle at e. A vertex v of P is mapped by G to a sphericalpolygon G(v), whose sides are the images under G of edges incident to v and whoseangles are easily seen to be the angles supplementary to the planar angles of thefaces incident to v; that is, G(e1) and G(e2) meet at angle π −α whenever e1 ande2 meet at angle α.

In other words, G(v) is exactly the “spherical polar” of the linkof v in P. (The link of a vertex is the intersection of a infinitesimal sphere centeredat v with P, rescaled, so that the radius is 1.)

4C. D. HODGSON, IGOR RIVIN, AND W. D. SMITHCollecting the above observations, it is seen that G(P) is combinatorially dualto P, while metrically it is the unit sphere S2.Now apply a similar construction to a convex polyhedron P in H3.

Associateto each vertex v of P a spherical polygon G(v) spherically polar to the link of v inP. Glue the resulting polygons together into a closed surface, using the rule thatG(v1) and G(v2) are identified isometrically whenever v1 and v2 share an edge.The resulting metric space G(P) is topologically S2 and the complex is stillPoincar´e dual to P.Metrically, however, it is no longer the round sphere.Tosee this, consider G(f)—the single common point of the spherical polygons G(vi),where vi is a vertex of f.The angle of G(vi) incident to G(f) is the exteriorangle of f at vi, and so by the Gauss-Bonnet Theorem, the sum of these angles is2π + area(f) ̸= 2π.

Thus G(f) is a cone-like singularity, or a cone point, with coneangle greater than 2π. (A cone angle equal to 2π corresponds to a smooth point.

)This analogue of the Gauss map turns out to have rather remarkable properties.Here is a brief summary:1. The image of a convex Euclidean polyhedron under the Gauss map is alwaysthe round sphere S2.

In sharp contrast, the following theorem holds.Theorem 4 (Compact Uniqueness). The metric of G(P) determines the hyper-bolic polyhedron P uniquely (up to congruence).The proof of uniqueness follows the argument used by Cauchy in the proof of hiscelebrated rigidity theorem for convex polyhedra in E3 (see [8, 4, or 24]).2.

Using the hyperboloid model of hyperbolic 3-space we can construct a model ofthe map G, which is not unlike the well-known spherical polar map. Let E31 denoteMinkowski space: R4 equipped with the inner product of signature −, +, +, +.Then H3 is represented by one sheet of the hyperboloid {x ∈E31 | ⟨x, x⟩= −1},which is the “sphere of radius √−1 ” in E31.

(For a thorough discussion of thehyperboloid model of H3 see [25, 7]. )The polar P ∗of a convex polyhedron P ⊂H3 consists of all outward Minkowskiunit normals to the support planes of P. Each such unit normal vector gives apoint in the the de Sitter Sphere S21 = {x ∈E31 | ⟨x, x⟩= 1}, which is the “sphereof radius 1” in E31.

It turns out that P ∗is a convex polyhedron in S21 and that theintrinsic metric of P ∗is exactly G(P).Note. The de Sitter sphere S21 is a semi-Riemannian submanifold of E31 of constantsectional curvature 1.

See [16] for further discussion of the geometry of E31 andsemi-Riemannian manifolds in general.3. We obtain a precise intrinsic characterization of those surfaces that can ariseas G(P) for a compact convex polyhedron P in H3.

The characterization is quiteeasy to state:Theorem 5. Characterization Theorem for compact polyhedra.

A met-ric space (M, g) homeomorphic to S2 can arise as the Gaussian image G(P) of acompact convex polyhedron P in H3 if and only if the following conditions hold:(a) The metric g has constant curvature 1 away from a finite collection ofcone points ci. (b) The cone angles at the ci are greater than 2π.

(c) The lengths of closed geodesics of (M, g) are all strictly greater than 2π.

CONVEX HYPERBOLIC AND CONVEX POLYHEDRA IN THE SPHERE5The necessity of (a) and (b) is immediately apparent from the above discussionof G. The necessity of (c) is based on hyperbolic version of Fenchel’s theorem (“thetotal geodesic curvature of a hyperbolic space curve is greater than 2π”) and the“polarity” model of the map G sketched in 2. See [21] for the details.The proof of the sufficiency of conditions (a)–(c) is based on Aleksandrov’s In-variance of Domain Principle (see [2, 4]), which exploits the observation that anopen and closed continuous map f from a topological space A into a connectedtopological space B is necessarily onto.Using this idea to prove Theorem 5 requires a careful study of the space Mn ofmetrics on S2 with n cone points satisfying conditions (a)–(c), of the space Pn ofconvex polyhedra in H3 with n faces, and of the Gauss map G : Pn →Mn.1.2.

Ideal polyhedra. The theory of the previous section is extended to noncom-pact polyhedra in [22].

Ideal polyhedra can be viewed as “boundary points” of Pn,and likewise Theorem 1 can be viewed as a “limiting case” of Theorem 5. In par-ticular, a polyhedral graph P ∗as in the statement of Theorem 1 can be completedto a piecewise-spherical metric on S2 by gluing in a standard round hemi-sphereinto each face.

It may be shown that this metric satisfies the conditions (a)–(c)of Theorem 5, except that it contains closed geodesics of length 2π, correspondingprecisely to the equators of the added hemi-spheres.Note. In [17] the necessity of the conditions of Theorem 1 is established withoutreference to the characterization of compact polyhedra.The techniques used to prove Theorem 5 are extended to prove Theorem 1 in[22].

The proof involves geometric estimates on families of convex polyhedra in H3whose vertices move away to the ideal boundary of H3 and beyond. The methodsactually suffice to produce a characterization of polyhedra of finite volume in H3,which includes Theorem 1 and Theorem 5 as special cases.

The techniques used toprove Theorem 4 give only partial uniqueness results for ideal polyhedra (see [17]).That approach also yields an algorithm for actually producing an ideal polyhedronin H3 with prescribed dihedral angles, which runs in time polynomial in the numberof vertices of the polyhedron and the number of decimals of accuracy required. Inother words this algorithm produces coordinates for a convex inscription of a graphinto the unit sphere in E3.

This is worthy of note, as the isometric embeddingresults of Aleksandrov et al. and Theorem 5 do not give an effective way to producea polyhedron with the desired properties.2.

AcknowledgmentsThe authors would like to thank Brian Bowditch and Mike Dillencourt for helpfuldiscussions. Igor Rivin would like to thank Bill Thurston.

He would also like tothank the NEC Research Institute for its hospitality, which made much of this workpossible.References1. A. V. Aho, J. E. Hopcroft, and J. D. Ullman, The design and analysis of computer algorithms,Addison-Wesley, Reading, MA, 1974.2.

A. D. Aleksandrov, An application of the theorem of invariance of domain to existence proofs,Izv. Akad.

Nauk SSSR Sci. Mat.

3 (1939), 243–255. (Russian; English Summary)

6C. D. HODGSON, IGOR RIVIN, AND W. D. SMITH3.

A. D. Aleksandrov, The intrinsic metric of a convex surface in a space of constant curvature,Dokl. Acad.

Sci. SSSR 45 (1944), 3–6.4.

A. D. Aleksandrov, Convex polyhedra, GITTL, Moscow, 1950. (Russian); German transl.

inAkademie Verlag Berlin 1958. MR 12, 732; 19, 1192.5.

E. M. Andreev, On convex polyhedra in Lobachevskii space, Math. USSR Sb.

10 (1970),413–440.6. E. M. Andreev, On convex polyhedra of finite volume in Lobachevskii space, Math.

USSR Sb.12 (1970), 255–259.7. Alan F. Beardon, The geometry of discrete groups, Springer-Verlag, New York, 1983.8.

A. L. Cauchy, Sur les polygones et poly`edres, 2nd memoir, J. ´Ecole Polytech.

19 (1813), 87–98.9. H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved problems in geometry, Springer-Verlag,1991.10.

M. Dillencourt and Warren D. Smith, Graph-theoretic aspects of inscribability, in preparation.11. M. B. Dillencourt, Toughness and Delaunay triangulations, J. Discrete Comput.

Geom. 5(1990), 575–601.12.

P. J. Federico, Descartes on polyhedra: A study of De Solidorum Elementis, Sources in theHistory of Mathematics and Physical Sciences, vol. 4, Springer-Verlag, New York, 1982.13.

M. Grotschel, L. Lovasz, and A. Schrijver, The ellipsoid method and its consequences incombinatorial optimization, Combinatorica 1 (1981), 169–197.14. Branko Gr¨unbaum, Convex polytopes, Wiley, New York, 1967.15.

C. D. Hodgson, Deduction of Andreev’s theorem from Rivin’s characterization of convexhyperbolic polyhedra, Topology 90, Proceeedings of the Research Semester in Low DimensionalTopology at O.S.U., de Gruyter Verlag (to appear).16. Barrett O’Neill, Semi-Riemannian geometry ; with applications to relativity, Academic Press,New York, 1983.17.

Igor Rivin, On geometry of convex ideal polyhedra in hyperbolic 3-space, Topology (to appear).18. Igor Rivin, On geometry of convex polyhedra in hyperbolic 3-space, PhD thesis, PrincetonUniv., June 1986.19.

Igor Rivin, Intrinsic geometry of convex polyhedra in hyperbolic 3-space, submitted.20. Igor Rivin, Some applications of the hyperbolic volume formula of Lobachevsky and Milnor,submitted.21.

Igor Rivin and C. D. Hodgson, A characterization of compact convex polyhedra in hyperbolic3-space, Invent. Math.

(to appear).22. Igor Rivin, A characterization of ideal polyhedra in hyperbolic 3-space, preprint 1992.23.

Jakob Steiner, Systematische Entwicklung der Abh¨angigkeit geometrischer Gestalten voneinander, Reimer, Berlin, 1832; Appeared in J. Steiner’s Collected Works, 1881.24. J. J. Stoker, Geometric problems concerning polyhedra in the large, Comm.

Pure AppliedMath. 21 (1968), 119–168.25.

William P. Thurston, Geometry and topology of 3-manifolds, Lecture notes, Princeton Univ.,1978.26. Pravin M. Vaidya, A new algorithm for minimizing convex functions over convex sets, IEEESympos.

Foundations of Computer Science, October 1989, pp. 338–343.Mathematics Department, University of Melbourne, Parkville, Victoria, AustraliaNEC Research Institute, Princeton, New Jersey 08540 and Mathematics Department,Princeton University, Princeton, New Jersey 08540NEC Research Institute, Princeton, New Jersey 08540

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