''Convex Polytopes'' is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra. It was written by

Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentVictor Klee,

Micha Perles Micah (; ) is a given name. Micah is the name of several people in the Hebrew Bible (Old Testament), and means "Who is like God?" The name is sometimes found with theophoric extensions. Suffix theophory in ''Yah'' and in '' Yahweh'' results in ...

, and

G. C. Shephard Geoffrey Colin Shephard is a mathematician who works on convex geometry and reflection groups. He asked Shephard's problem on the volumes of projected convex bodies, posed another problem on polyhedral nets, proved the Shephard–Todd theorem in ...

, and published in 1967 by John Wiley & Sons. It went out of print in 1970. A second edition, prepared with the assistance of Volker Kaibel, Victor Klee, and

Günter M. Ziegler Günter Matthias Ziegler (born 19 May 1963) is a German mathematician who has been serving as president of the Free University of Berlin since 2018. Ziegler is known for his research in discrete mathematics and geometry, and particularly on the ...

, was published by

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 ...

in 2003, as volume 221 of their book series

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) ( ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standa ...

. ''Convex Polytopes'' was the winner of the 2005 Leroy P. Steele Prize for mathematical exposition, given by the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...

. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries.

Topics

The book has 19 chapters. After two chapters introducing background material in linear algebra, topology, and

convex geometry In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of n ...

, two more chapters provide basic definitions of polyhedra, in their two dual versions (intersections of half-spaces and convex hulls of finite point sets), introduce

Schlegel diagram In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the ori ...

s, and provide some basic examples including the cyclic polytopes. Chapter 5 introduces

Gale diagram In the mathematical discipline of polyhedral combinatorics, the Gale transform turns the vertices of any convex polytope into a set of vectors or points in a space of a different dimension, the Gale diagram of the polytope. It can be used to descr ...

s, and the next two chapters use them to study polytopes with a number of vertices only slightly higher than their dimension, and neighborly polytopes. Chapters 8 through 11 study the numbers of faces of different dimensions in polytopes through Euler's polyhedral formula, the Dehn–Sommerville equations, and the extremal combinatorics of numbers of faces in polytopes. Chapter 11 connects the low-dimensional faces together into the

skeleton A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...

of a polytope, and proves the van Kampen–Flores theorem about non-embeddability of skeletons into lower-dimensional spaces. Chapter 12 studies the question of when a skeleton uniquely determines the higher-dimensional combinatorial structure of its polytope. Chapter 13 provides a complete answer to this theorem for three-dimensional convex polytopes via Steinitz's theorem, which characterizes the graphs of convex polyhedra combinatorially and can be used to show that they can only be realized as a convex polyhedron in one way. It also touches on the multisets of face sizes that can be realized as polyhedra (

Eberhard's theorem In mathematics, and more particularly in polyhedral combinatorics, Eberhard's theorem partially characterizes the multisets of polygons that can form the faces of simple convex polyhedra. It states that, for given numbers of triangles, quadrilateral ...

) and on the combinatorial types of polyhedra that can have inscribed spheres or

circumscribed sphere In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle ...

s. Chapter 14 concerns relations analogous to the Dehn–Sommerville equations for sums of angles of polytopes, and uses sums of angles to define a central point, the "Steiner point", for any polytope. Chapter 15 studies

Minkowski addition In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski ...

and Blaschke addition, two operations by which polytopes can be combined to produce other polytopes. Chapters 16 and 17 study shortest paths and the

Hirsch conjecture In mathematical programming and polyhedral combinatorics, the Hirsch conjecture is the statement that the edge-vertex graph of an ''n''-facet polytope in ''d''-dimensional Euclidean space has diameter no more than ''n'' − ''d''. That is ...

, longest paths and

Hamiltonian cycle In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex ...

s, and the

shortness exponent In graph theory, the shortness exponent is a numerical parameter of a family of graphs that measures how far from Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian ...

of polytopes. Chapter 18 studies

arrangements of hyperplanes In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set ''A'' of hyperplanes in a linear, affine, or projective space ''S''. Questions about a hyperplane arrangement ''A'' generally concern geometrical, top ...

and their dual relation to the combinatorial structure of zonotopes. A concluding chapter, chapter 19, also includes material on the symmetries of polytopes. Exercises throughout the book make it usable as a textbook, and provide additional links to recent research, and the later chapters of the book also list many open research problems. The second edition of the book keeps the content, organization, and pagination of the first edition intact, adding to it notes at the ends of each chapter on updates to the material in that chapter. These updates include material on

Mnëv's universality theorem In algebraic geometry, Mnëv's universality theorem is a result which can be used to represent algebraic (or semi algebraic) varieties as realizations of oriented matroids, a notion of combinatorics. Oriented matroids For the purposes of Mnëv's ...

and its relation to the realizability of polytopes from their combinatorial structures, the proof of the

g

-conjecture for simplicial spheres, and Kalai's conjecture. The second edition also provides an improved bibliography. Topics that are important to the theory of convex polytopes but not well-covered in the book ''Convex Polytopes'' include

Hilbert's third problem The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely ...

and the theory of Dehn invariants.

Audience and reception

Although written at a graduate level, the main prerequisites for reading the book are

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...

and

general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...

, both at an undergraduate level. In a review of the first edition of the book, Werner Fenchel calls it "a remarkable achievement", "a wealth of material", "well organized and presented in a lucid style". Over 35 years later, in giving the Steele Prize to Grünbaum for ''Convex Polytopes'', the American Mathematical Society wrote that the book "has served both as a standard reference and as an inspiration", that it was in large part responsible for the vibrant ongoing research in

polyhedral combinatorics Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral c ...

, and that it remained relevant to this area. Reviewing and welcoming the second edition, Peter McMullen wrote that despite being "immediately rendered obsolete" by the research that it sparked, the book was still essential reading for researchers in this area.

References

{{reflist, refs= {{citation , last = Baxandall , first = P. R. , date = October 1969 , doi = 10.2307/3615008 , issue = 385 , journal =

The Mathematical Gazette ''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...

, pages = 342–343 , title = Review of ''Convex Polytopes'' (1st ed.) , volume = 53 {{citation, title=''Convex Polytopes'' (Basic Library List selection, no review), work=MAA Reviews, publisher= Mathematical Association of America, accessdate=2020-08-26, url=https://www.maa.org/press/maa-reviews/convex-polytopes-0 {{citation , last = Ehrig , first = G. , language = German , title = Review of ''Convex Polytopes'' (2nd ed.) , work = zbMATH , zbl = 1024.52001 {{citation , last = Fenchel , first = Werner , authorlink = Werner Fenchel , date = Winter 1968 , issue = 4 , journal = American Scientist , jstor = 27828384 , pages = 476A–477A , title = Review of ''Convex Polytopes'' (1st ed.) , volume = 56 {{citation , last = Jucovič , first = E. , language = German , title = Review of ''Convex Polytopes'' (1st ed.) , work = zbMATH , zbl = 0163.16603 {{citation , last = Lord , first = Nick , date = March 2005 , issue = 514 , journal =

, jstor = 3620690 , pages = 164–166 , title = Review of ''Convex Polytopes'' (2nd ed.) , volume = 89 {{citation , last = McMullen , first = Peter , authorlink = Peter McMullen , date = July 2005 , doi = 10.1017/s0963548305226998 , issue = 4 , journal =

Combinatorics, Probability and Computing ''Combinatorics, Probability and Computing'' is a peer-reviewed scientific journal in mathematics published by Cambridge University Press. Its editor-in-chief is Béla Bollobás (DPMMS and University of Memphis). History The journal was establ ...

, pages = 623–626 , title = Review of ''Convex Polytopes'' (2nd ed.) , volume = 14 {{citation , last = Sallee , first = G. T. , mr = 0226496 , title = Review of ''Convex Polytopes'' (1st ed.) , work =

MathSciNet MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal '' Mathematical Reviews'' (MR) since 1940 along with an extensive author database, links ...

{{citation , date = April 2005 , issue = 4 , journal = Notices of the American Mathematical Society , pages = 439–442 , title = 2005 Steele Prizes , url = https://www.ams.org/notices/200504/comm-steele.pdf , volume = 52 {{citation , last = Zvonkin , first = Alexander , mr = 1976856 , title = Review of ''Convex Polytopes'' (2nd ed.) , work =

, year = 2004 Polytopes Convex geometry Mathematics textbooks 1967 non-fiction books 2003 non-fiction books

Topics

Audience and reception

See also

References