Approaching convex and discrete geometry from other disciplines

Question

I would like to learn some convex and discrete geometry (number 52 in MSC2010). I thought that it would be interesting to approach it from some other parts of mathematics - either by learning applications of convex and discrete geometry to these parts or (preferably) the other way round.

Question: What are some interesting, non-trivial, preferably "important" results or problems lying in the intersection of convex and discrete geometry with other parts of mathematics?

I am most interested in "other parts" being topology (including algebraic and geometric topology), order theory, logic (including model theory). However, other disciplines are also welcome.

I did some MathSciNet searches, but I am not sure which papers could be interesting.

Answer 1

Here is a list of applications of other kinds of math to discrete and convex geometry (there is a little overlap with other answer already given). Some are not as important or groundbreaking as others but I consider all of them as reasonably interesting. The list comes in no particular order.

1. Applications of topology to discrete geometry aka "Topological Combinatorics" -- Examples include the Ham Sandwich Theorem for measures, the colorful Tverberg theorem, and many more. A good starting point is the book "Using the Borsuk-Ulam theorem" by J. Matousek.
2. Upper bound theorem for spheres and the $g$-conjecture -- This is a breathtaking application of commutative algebra and algebraic geometry to convex polytopes (and more generally spheres). If you want to learn from the master himself, have a look at Stanley's book "Combinatorics and commutative algebra".
3. Polynomial method -- Using this (somewhat) recently developed technique, Guth and Katz were able to almost prove the conjectured lower bound of the Erdös distance problem. A good starting point to learn about this might be the blog entry by Tao.
4. Monsky's theorem -- Monsky's theorem says that you cannot dissect a square in an odd number of triangles with all triangles having the same area. The proof makes essential use of 2-adic valuations and Sperner's lemma from topology. For this and many more results have a look at the book "Algebra and Tiling: Homomorphisms in the Service of Geometry" by Stein & Szabo. Monsky's result is also covered in one of the more recent editions of "Proofs from the book" by Aigner & Ziegler.
5. The Kalai-Kleitman bound on the diameter of the graph of a polytope -- One of the biggest open problems in the theory of polytopes is the question of how large the diameter of the graph of a convex $d$-polytope with $n$ facets can be. Kalai & Kleitman showed that the diameter is at most $n^{\log(d) + 2}$. Very recently Todd was able to improve this bound to $(n-d)^{\log d}$. The technqiues used stem from extremal (hyper)graph theory and are not difficult to understand.
6. Applications of differential geometry to discrete geometry -- Adiprasito's thesis contains nice examples of such applications. In particular he (together with Ziegler) was able to prove that in high enough dimension there are infinitely many projectively unique polytopes (which kills a conjecture by Shephard & McMullen).
7. Optimization applied to discrete geometry -- de Oliveira Filho & Vallentin applied semi-definite programming to study ball packings and the chromatic number of $\mathbb{R}^d$. Have also a look at Firsching's paper Computing maximal copies of polytopes contained in a polytope which is another nice application of optimization techniques.
8. Stochastic geometry -- Closely related to convex geometry is stochastic geometry. Schneider & Weil wrote a book on the topic.
9. An application of the Möbius function from order theory to hyperplane arrangements -- This application is nicely describe in the new edition of Stanley's "Enumerative Combinatorics, vol I" (see Section 3.11).
10. Eulerian posets -- These are special posets that have a lot of nice properties; a very good introduction is again Stanley's EC1. In particular, every face lattice of a polytope is a eulerian poset. This way we know that the Generalized Dehn-Sommerville equations for polytopes hold true, see the article by Bayer & Billera.

Answer 2

1) Theory of toric varieties is a link between convex geometry and algebraic geometry. To any convex $n$-dimensional polytope in $\mathbb{R}^n$ one constructs a complex projective variety of complex dimension $n$ with action of the complex torus $(\mathbb{C}^*)^n$. This action has finitely many orbits, in fact as many as there are faces of the polytope. Moreover one can establish 1-1 correspondence between faces and orbits. It is established via the so called moment map which maps the toric variety to the polytope. Based on this connection, there is a number of results obtained about combinatorics of polytopes using the methods of algebraic geometry, e.g. McMullen's conjecture on h-vector proved by Stanley. There is a book on this and other material "Convex bodies and algebraic geometry" by T. Oda.

2) Theory of valuations on convex sets studies finitely additive measures on the class of convex sets, usually with some kind of continuity properties. It has traditionally strong relations to integral geometry. Recently classification of unitarily invariant valuations has been applied to integral geometry of $\mathbb{C}^n$ (kinematic formulas); see http://arxiv.org/abs/0801.0711 .

Answer 3

For convex geometry & functional analysis:

Ball, Keith Convex geometry and functional analysis. Handbook of the geometry of Banach spaces, Vol. I, 161–194, North-Holland, Amsterdam, 2001.

Ball, Keith An elementary introduction to modern convex geometry. Flavors of geometry, 1–58, Math. Sci. Res. Inst. Publ., 31, Cambridge Univ. Press, Cambridge, 1997.

For discrete geometry & functional analysis:

Ostrovskii, Mikhail I.(1-STJ) Metric embeddings. Bilipschitz and coarse embeddings into Banach spaces. De Gruyter Studies in Mathematics, 49. De Gruyter, Berlin, 2013.

Answer 4

Since you mention topology... One tiny corner of knot theory concerns knots drawn on $\mathbb{Z}^3$. In this MO answer, Ryan Budney identifies these as plumber's knots. There is quite a bit of work on modeling polymer chains in 3D lattices, e.g., this paper on "Under-knotted and over-knotted polymers."

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Some questions:

Q. What is the minimum number of segments ("sticks") in a drawing on $\mathbb{Z}^3$ of a given knot $K$? How difficult is it (computationally) to find such a minimal stick lattice realization of $K$, say, as a function of the crossing number of $K$?

Answer 5

One just has to say:   Geometry of Numbers.

See C.G.Lekkerkerker, Geometry of numbers, SBN: 7204 2108 X (right, this book lists SBN, not ISBN).