Convex Polyhedra by A. D. AlexandrovReview by: Robert ConnellySIAM Review, Vol. 48, No. 1 (Mar., 2006), pp. 157-160Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/20453762 .

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BOOK REVIEWS 157

Bubnov-Galerkin approach again. As re gards uniqueness, it is established under some assumptions that are not made com pletely clear (in particular-but not only due to typographical errors in equation ref erences).

Chapter 6 gives the formulation of a phys ically nonlinear problem choosing a specific nonlinear constitutive equation, and Chap ter 7 (more than 100 pages long!) presents a collection of numerical results for this type of model in various configurations using fi nite difference discretizations. The range of problems considered is indeed quite im pressive, but unfortunately no experimental comparison is given. Then Chapter 8 ex tends the formulation to include plasticity effects, with various numerical solutions.

Finally, Chapter 9 sets out to provide a list of "The problems in the field of non linear dynamics of shells that remain to be solved..." I

As a conclusion, this book can certainly be useful to shell experts, especially as re gards the techniques demonstrated to model and mathematically analyze thermoelastic effects in shell structures. However, I can not recommend it to a wider audience, in particular due to the lack of up-to-date nu merical procedures (obviously, shell struc tures are most effectively analyzed using finite element procedures) and of experi mental confrontation or assessment of any kind for the complex modeling considered.

DOMINIQUE CHAPELLE INRIA, France

Convex Polyhedra. By A.D. Alexandrov. Sprin ger-Verlag, Berlin, 2005. $129.00. xii+539 pp., hardcover. ISBN 3-540-23158-7.

This book was first published in Russian in 1950 [2], then translated into German in 1958 [3]. Now, V. A. Zalgaller has updated and substantially lengthened this English edition with more recent related results and translations of hard-to-find related results. Nevertheless, the earlier Russian and Ger man versions of this book were part of a sig nificant contribution to research concerning the geometry of polyhedra, especially con vex polyhedra in three-space, despite their,

until now, lack of accessibility. (It seems that there are only two copies of the origi nal Russian edition available in the United States through the interlibrary loan sys tem.) It is a mystery to me why such a trans lation into English was not done long ago.

For me, the star result in this book has to do with the realizability of developments of convex polyhedra. Suppose you have a com pact, convex (bounded) three-dimensional polyhedron P sitting on your table made out of cardboard. Take a knife and slit it open in such a way that the resulting cardboard lies flat on the table as one con nected piece, possibly with overlaps. This is a development. It can be thought of as a flat polygonal disk represented in the plane, with identifications on its boundary to de termine how it fits onto the surface of P. The development is essentially just the in trinsic metric surface of P.

If you look at the development in a neigh borhood of what was a vertex v of P, the sum s of the internal angles at v of the faces of P will be strictly less than 27r, and 27r-s is the intrinsic curvature of P at v. It is a basic result that the sum of the curvatures of all the vertices of P is 4wr. The convexity of P implies that this intrinsic curvature at each vertex is positive, and for all the other points in the development, the curvature is 0.

Suppose you start with a development, thought of as a metric disk with identifi cations, such that the intrinsic curvature at each point is nonnegative, only a finite number are positive, and the total curvature is 4wr. So, topologically, you get a surface homeomorphic to a two-dimensional sphere. One of the most important results explained in the book is that for each such intrinsi cally convex development, there is a unique (up to global congruence) convex polyhe dron P having that development, with the possible degenerate case when P becomes a doubly covered convex planar polygon. This is Alexandrov's existence theorem. It is explained carefully in this book and is a substantial accomplishment.

After Alexandrov's first proof of this result appeared in the 1940s [1], A. V. Pogorelov [10] generalized it considerably to the case where the surface was any sort of metric space homeomorphic to a two dimensional sphere but still intrinsically

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158 BOOK REVIEWS

convex, while at the same time extending the definition of intrinsic convexity to allow for the boundary of any convex set in three space. Pogorelov's result was also described in a book by him that was translated into English [10] in the 1950s. But Pogorelov's proof was quite complicated and lengthy. In the present translation of A. D. Alexan drov's book, in addition to several footnotes concerning more recent results, Zalgaller has included supplements which are transla tions of papers of Yu. A. Volkov that explain a somewhat simpler and shorter (but still quite nontrivial) proof of Pogorelov's result above. (But note that the paper [13] gives a somewhat more general treatment than the one translated in the book reviewed here.) This gives the book a completeness and ac cessibility that has so far been sadly lacking, at least for English speakers in the West.

Existence and uniqueness results, espe cially for polyhedra, have a long and dis tinguished history going back at least to Cauchy in 1813 and even, in spirit, to Eu clid's Elements. Cauchy showed that if two convex polyhedra P and Q in three-space are such that there is a continuous cor respondence from the surface of P to the surface of Q, which restricted to each face of P and Q is a rigid congruence, then P and Q are congruent (allowing for the pos sibility of reflected images). This is one of the first uniqueness (or rigidity) theorems. Cauchy's proof had some hiccups that were not noticed for some time and eventually repaired [12], but the basic ideas were very insightful and still form a foundation for the theory of rigid polyhedra.

There are at least two points of view con cerning such existence and rigidity results for polyhedra. One is to emphasize the sur face of the polyhedron as a metric manifold as if it were a piece of paper. The other is to regard the vertices as universal joints and the edges as fixed-length bars connect ing those joints, thinking of the polyhedron as a framework, giving the subject a more discrete and combinatorial flavor. The Rus sian school with A. D. Alexandrov, V. A. Pogorelov, N. V. Efimov, and many oth ers tended toward the first viewpoint, while some others such as W. Whiteley, H. Crapo, and myself, while coming much later to the subject, have tended to the more discrete

viewpoint. But both viewpoints are quite compatible, where they overlap.

In 1916, M. Dehn [7] proved an anal ogous result to Cauchy's for infinitesimal deformations of convex polyhedra in three space. An infinitesimal deformation (or in finritesimal flex) of a polyhedron is a vector field associated to the polyhedron that is trivial when restricted to each face. (A vec tor field being trivial means that it extends to the derivative of a differentiable family of congruences, starting at the identity, of all of Euclidean space at time 0.) Dehn's result says that any infinitesimal flex of a convex polyhedron in three-space is glob ally trivial. In a formal sense, this proof is somewhat simpler than Cauchy's and is nicely described in Alexandrov's book. In deed, in [8] H. Gluck describes Alexandrov's proof of Dehn's theorem, while he observes that from the point of view of frameworks, almost all (in the measure theoretic sense, for example) triangulated spheres in three space are rigid. Gluck also repeated the rigidity conjecture that says that any em bedded triangulated surface in three-space is rigid, using this result as "evidence" for it. This paper was the starting point for

my interest in this subject, which led to my discovery of a flexible embedded tri angulated sphere, a counterexample to the rigidity conjecture [6]. Interestingly, also, in the category of convex polyhedral metrics, Olovyanishnikov [11] showed that if two convex polyhedral surfaces are isometric as surfaces, then the polyhedra themselves are congruent.

Briefly, for his existence result, Alexan drov first extends the Cauchy-Dehn-Olov yanishnikov uniqueness result about poly hedra in three-space, where (infinitesimal) foldings are allowed across faces and also the possibility that the convex polytope could become nonconvex by some small defor mation. There is a natural map from the space of convex polyhedra to the space of convex polyhedral metrics that is essentially locally one-to-one by the uniqueness prop erty. Then his key new idea was to apply the topological invariance of domain to show that the map is onto. Both of these steps are tricky and need to be done carefully.

But Alexandrov's book has several inter esting discussions of other closely related

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BOOK REVIEWS 159

subjects in addition to the existence and uniqueness of polyhedra having given met rics. For example, a theorem of Minkowski says that if unit vectors in three-space and corresponding positive scalars are given, the sum of the scalars times the correspond ing normal vectors being zero is neces sary and sufficient for the existence of a convex polytope in three-space whose face normals are the given vectors and whose corresponding areas are the given scalars. Furthermore, it is shown how the poly tope that realizes these normal vectors and face areas is unique. Alexandrov extends

Minkowskii's theorem to show that if there are two polytopes with the same normals and such that corresponding faces do not fit inside each other, then they are con gruent. A corollary is that if two poly hedra have the same face normals and face perimeters, then they are congruent. All this is done using very similar argu ments as with the realization problem of polyhedral metrics. In all these problems, there is a lot of discussion about the sit uation for unbounded polyhedra and poly hedra with boundaries, situations that are complicated and need to be handled with care. Another interesting topic is Alexan drov's discussion of the realization of con vex polyhedra whose vertices lie on rays from a point. There is also a very interest ing discussion connecting the Cauchy-type problem with the Minkowski-type prob lem formally and showing how the Brunn

Minkowski inequalities can be brought to bear.

I would definitely recommend this book to a student who would like to get ac quainted with some of the ideas in this sort of discrete geometry. There are little goodies throughout that are very enlight ening, and the discussion is very conver sational. Unfortunately, though, there are few exercises, and complicated extensions are treated along with the basic more inter esting ones without pause. In this English translation under review, there are a few ty pos here and there, such as the statement in the introduction that the date of Cauchy's rigidity result was 1913 instead of 1813, but generally the text is quite readable. There are treatments of some of Alexandrov's re sults in English such as in the little book [9],

for example, but Alexandrov's existence re sult is not accessible there, and Lyusternik's description of Steinitz's theorem, about the realization of convex polyhedra with a given combinatorial type, is faulty.

In the interests of disclosure, I should mention that I was asked by Springer to help oversee an earlier abortive attempt at an English translation of this book, but I was not involved with this final translation.

It is also interesting that A. D. Alexan drov himself had a long and distinguished career, receiving several prizes such as the Stalin prize for his solution of the Herman Weyl problem in 1942, the Lobachevsky prize for his geometric results in 1951, and being elected a full member of the Academy of Sciences of the USSR in 1964. In addi tion, he was the Rector of Leningrad Uni versity from 1952 to 1964. He did his semi nal work during the unimaginably difficult times leading up to and during World War II and under Stalin. Nevertheless, his work, especially the work represented in this book, had a great influence on countless Russian mathematicians such as A. V. Pogorelov, M. Gromov, I. Pak, I. Rivin, and V. A. Zalgaller, as well as those of us outside of Russia, such as H. Gluck, B. Roth, and L. Asimow (see [4] and [5]).

I thank I. Pak, I. Rivin, and N. Dolbilin for many very helpful comments and much helpful information.

This brings us back to the question of why there was such neglect in translating this book into English. I don't know.

REFERENCES

[1] A. D. ALEXANDROV, Existence of a convex

polyhedron and a convex surface with a given metric, Mat. Sb., 11 (1942), pp. 1-2, 15-65 (in Russian).

[2] A. D. ALEXANDROV, Vypuklye mno gogranniki [Convex Polyhedra], Go sudarstv. Jzdat. Tehn.-Teor. Lit., Moscow, Leningrad, 1950.

[3] A. D. ALEXANDROW, Konvexe polyeder, Mathematiksche Lehrbiicher und Monographien. Herausgegeben von der Deutschen Akademie der Wis

senschaften zu Berlin, Forschungsin stitut fur Mathematik. II. Abteilung: Mathematische Monographien, Bd. VIII. Akademie-Verlag, Berlin, 1958.

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160 BOOK REVIEWS

[4] L. ASIMOW AND B. ROTH, The rigidity of graphs, Trans. Amer. Math. Soc., 245 (1978), pp. 279-289.

[5] L. ASIMOW AND B. ROTH, The rigidity of

graphs. II, J. Math. Anal. Appl., 68

(1979), pp. 171-190.

[6] R. CONNELLY, The rigidity of polyhe dral surfaces, Math. Mag., 52 (1979),

pp. 275-283. [7] M. DEHN, Uber die Starheit konvexer

polyeder, Math. Ann., 77 (1916), pp. 466-473.

[8] H. GLUCK, Almost all simply connected closed surfaces are rigid, in Geo

metric Topology (Proc. Conf., Park

City, Utah, 1974), Lecture Notes in

Math. 438, Springer-Verlag, Berlin,

1975, pp. 225-239.

[9] L. A. LYUSTERNIK, Convex Figures and Polyhedra, translated from the Rus sian by T. J. Smith, Dover, New York,

1963. [10] A. V. POGORELOV, Extrinsic Geometry

of Convex Surfaces, translated from the Russian by Israel Program for

Scientific Translations, Transl. Math.

Monogr. 35, AMS, Providence, RI,

1973. [11] S. P. OLOVYANISHNIKOV, A generaliza

tion of Cauchy's theorem about con

vex polyhedra, Mat. Sb., 18 (1946), pp. 441-446 (in Russian).

[12] E. STEINITZ, PolyederundRaumeinteilun gen, in Encycl. Math. Wiss. III, 12,

Leipzig, 1922, pp. 1-139. [13] Ju. A. VOLKOV AND E. G. PODGOR

NOVA, Existence of a convex poly hedron with a given evolute, Ge

ometriceskiY Sbornik. Taskent. Gos. Ped. Inst. Ucen. Zap., 85 (1971),

pp. 3-54 (in Russian).

ROBERT CONNELLY

Cornell University

An Introduction to Nonlinear Analysis. By Martin Schechter. Cambridge University Press, Cambridge, UK, 2005. $75.00. xviii+357 pp., hardcover. ISBN 0-521-84397-9.

The central idea in this book is to give an introduction to nonlinear analysis without becoming too involved in the technicali ties. To achieve this, in most of the book, Schechter restricts himself to applications in ordinary differential equations. This is an interesting idea and seems to work very well.

A second idea in the book is to introduce mathematical problems to motivate theory.

This also seems to work well. In general, I find the style pleasant and highly readable. There are good sets of exercises.

Schechter introduces many very useful techniques in the book, such as mollifiers, weak derivatives, and basic calculus of vari ations. In addition, he briefly discusses a number of applications of the main ideas.

In the variational methods, the author discusses the usual topics, such as minima, mountain pass points, and saddle points, as well as jumping nonlinearities. At the end of the book, he does discuss the higher dimensional case.

Schechter has a number of appendices dis cussing material needed from preliminary courses.

The construction of the Brouwer degree in section 6.5 seems incomplete. In particu lar Schechter does not prove correctly that d(q, Q, p) is well defined if p is a singular value and q is C1. The difficulty is that the singular values of q may locally disconnect Dn

As a first course, I do feel it has some disadvantages. For my taste, there is an overemphasis on variational methods. Sec ond, I feel that a first course should contain some basic bifurcation theory. I stress that he does introduce degree theory.

This would make quite a nice book for a course, though I feel the user might want to supplement the material.

E. NORMAN DANCER University of Sydney

Ordered Sets. By Egbert Harzheim. Springer Verlag, New York, 2005. $129.00. xii+386 pp., hardcover. ISBN 0-387-24219-8.

Ordering is intrinsic to notions of magni tude, containment, and incidence. Thus, (partially) ordered sets are ubiquitous ob jects in diverse areas of mathematics, arising in algebra, combinatorics, geometry, logic, set theory, and topology. Abstraction of the axioms defining a partial order as a reflexive, antisymmetric, transitive binary relation, came at the end of the nineteenth century in work of Peirce, Schroder, and Dedekind, though systematic study of or dered sets began half a century before with Boole's investigations of the laws of thought. In this review, we adopt the terminology

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