''Convex Polytopes'' is a graduate-level mathematics textbook about

convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

s, higher-dimensional generalizations of three-dimensional

convex polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surfa ...

. It was written by

Branko Grünbaum Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentVictor Klee Victor LaRue Klee, Jr. (September 18, 1925 – August 17, 2007) was a mathematician specialising in convex sets, functional analysis, analysis of algorithms, optimization, and combinatorics. He spent almost his entire career at the University of ...

, Micha Perles, and G. C. Shephard, 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, 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 ...

in 2003, as volume 221 of their book series

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) () 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 standard size (with va ...

. ''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 The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university A university () is an educational institution, institution of tertiary edu ...

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 num ...

, two more chapters provide basic definitions of polyhedra, in their two dual versions (intersections of half-spaces and

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

s 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 polytope In mathematics, a cyclic polytope, denoted ''C''(''n'', ''d''), is a convex polytope formed as a convex hull of ''n'' distinct points on a rational normal curve in R''d'', where ''n'' is greater than ''d''. These polytopes were studied by Constanti ...

s. Chapter 5 introduces Gale diagrams, and the next two chapters use them to study polytopes with a number of vertices only slightly higher than their dimension, and

neighborly polytope In geometry and polyhedral combinatorics, a -neighborly polytope is a convex polytope in which every set of or fewer vertices forms a face. For instance, a 2-neighborly polytope is a polytope in which every pair of vertices is connected by an e ...

s. Chapters 8 through 11 study the numbers of faces of different dimensions in polytopes through Euler's polyhedral formula, the

Dehn–Sommerville equations In mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their gen ...

, 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 most animals. There are several types of skeletons, including the exoskeleton, which is a rigid outer shell that holds up an organism's shape; the endoskeleton, a rigid internal fra ...

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 In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedron, convex polyhedra: they are exactly the vertex connect ...

, 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) and on the combinatorial types of polyhedra that can have

inscribed sphere image:Circumcentre.svg, An inscribed triangle of a circle In geometry, an inscribed plane (geometry), planar shape or solid (geometry), solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figu ...

s 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 Vertex (geometry), vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the te ...

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 of two set (mathematics), sets of position vectors ''A'' and ''B'' in Euclidean space is formed by vector addition, adding each vector in ''A'' to each vector in ''B'': A + B = \ The Minkowski difference (also ''M ...

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

shortest paths In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between t ...

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 ...

, longest paths and

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

s, and the shortness exponent of polytopes. Chapter 18 studies arrangements of hyperplanes and their dual relation to the combinatorial structure of

zonotope In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...

s. 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 mathematics, Mnëv's universality theorem is a result in the intersection of combinatorics and algebraic geometry used to represent algebraic (or semialgebraic) varieties as realization spaces of oriented matroids. Informally it can also be u ...

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

g

-conjecture for

simplicial sphere In geometry and combinatorics, a simplicial (or combinatorial) ''d''-sphere is a simplicial complex homeomorphic to the ''d''-dimensional sphere. Some simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions mos ...

s, 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 problems, 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 polyhedron, polyhedra of equal volume, is it always possible t ...

and the theory of

Dehn invariant In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled (" dissected") into another, and whether a polyhedron or its dissections can tile space. It is named after Max Dehn, who ...

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 matrix (mathemat ...

and

general topology In mathematics, general topology (or point set 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 differ ...

, both at an undergraduate level. In a review of the first edition of the book,

Werner Fenchel Moritz Werner Fenchel (; 3 May 1905 – 24 January 1988) was a German-Danish mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear opti ...

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 co ...

, and that it remained relevant to this area. Reviewing and welcoming the second edition,

Peter McMullen Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London. Education and career McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at ...

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

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The Mathematical Gazette ''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journ ...

, 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=

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zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastru ...

, zbl = 1024.52001 {{citation , last = Fenchel , first = Werner , authorlink = Werner Fenchel , date = Winter 1968 , issue = 4 , journal =

American Scientist ''American Scientist'' (informally abbreviated ''AmSci'') is an American bimonthly science and technology magazine published since 1913 by Sigma Xi, The Scientific Research Honor Society. In the beginning of 2000s the headquarters was moved to ...

, 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 =

, zbl = 0163.16603 {{citation , last = Lord , first = Nick , date = March 2005 , issue = 514 , journal =

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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 estab ...

, 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 Graduate Texts in Mathematics 1967 non-fiction books 2003 non-fiction books Mathematics textbooks

Topics

Audience and reception

See also

References