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20 votes
5 answers
1k views

From convex polytopes to toric varieties: the constructions of Davis and Januszkiewicz

One of the most useful tools in the study of convex polytopes is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on ...
Gil Kalai's user avatar
  • 24.7k
19 votes
3 answers
2k views

Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?

Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...
Tom Copeland's user avatar
  • 10.5k
19 votes
2 answers
1k views

About a Delzant polytope. (In particular dodecahedron)

Hi. I have a question. Definition. Delzant polytope $P$ is a rational convex simple polytope with the smooth condition. Here, "smooth" means that for each vertex $v$, the $n$ edges containing $v$ ...
Yunhyung Cho's user avatar
  • 1,037
12 votes
2 answers
665 views

Detecting tilings by toric geometry

This is probably a silly question, but I figured that if there is a good answer, this would be a good place to ask. Ever since I got my hands on the book "Toric Varieties" by Cox, Little and Schenck, ...
Gjergji Zaimi's user avatar
10 votes
2 answers
224 views

The set of polytopes with given $f$-vector

Let $f=(f_0,\ldots f_n)$ be a vector in $\Bbb N^{n+1}$. Let $X$ be the set of all (ordered) $f_0$-tuples in $\Bbb R^n$ whose convex hull has $f$ as its $f$-vector. Assume that $X$ is non-empty. Is ...
Avi Steiner's user avatar
  • 3,079
9 votes
1 answer
1k views

Finding the vertices of a polyhedral complex coming from a GIT wall and chamber decomposition

I am interested in a polyhedral/combinatorics problem that arises in algebraic geometry in the context of geometric invariant theory (GIT). Algebro-geometric background: Consider the natural ...
Noah Giansiracusa's user avatar
6 votes
1 answer
539 views

Proofs of Euler's characteristic formula for n-Dim polytopes

Twenty proofs of Euler's formula V - E + F - 1 = 1, which applies to convex polyhedrons, i.e., 3-dimensional polytopes, are presented at the Geometry Junkyard. I'm interested in proofs of the more ...
Tom Copeland's user avatar
  • 10.5k
4 votes
1 answer
189 views

2-faces of reflexive Delzant polytopes

Question 1. Can a reflexive Delzant polytope of some dimension contain a $2$-face with more than $11$ edges? Motivation. I would like more generally to get an answer to the following question: ...
aglearner's user avatar
  • 14.3k
3 votes
1 answer
336 views

Cohomology ring of a hypersurface in toric variety

Let $X^n$ be a smooth projective toric variety corresponding to a simple polytope $P$. It is well known that the cohomology ring $H^*(X)$ can be described in terms of combinatorics of faces of $P$. ...
asv's user avatar
  • 21.8k
3 votes
0 answers
179 views

Polytope algebra and toric vareties

Let $\Pi$ denote the McMullen polytope algebra (over the field of rationals $\mathbb{Q}$) generated by convex polytopes with rational vertices in $\mathbb{R}^n$. For a simple polytope $P$ let us ...
asv's user avatar
  • 21.8k
3 votes
0 answers
102 views

The ring generated by a convex polytope and its faces

Let $V=\Bbb R^n$. Morelli defined the (commutative unital) ring $L(V)$ to be the additive group generated by the indicator functions of convex polytopes in $V$ with multiplication induced by Minkowski ...
Avi Steiner's user avatar
  • 3,079
2 votes
0 answers
94 views

Anything similar to cone product formula (for convex polytopes)?

The convex polytope flag vector ring $\mathcal{R}$ satisfies the cone product formula $$ C(U) C(V) = C(J(U, V)) + DUV $$ where $$ J(U, V) = U C(V) + C(U) V - e_1 UV $$ is the join formula. Note: ...
Jonathan Fine's user avatar
0 votes
0 answers
42 views

When is the set of faces of a convex polytope algebraically independent?

This is related to another question of mine Let $V=\Bbb R^n$. Morelli defined the (commutative unital) ring $L(V)$ to be the additive group generated by the indicator functions of convex polytopes in ...
Avi Steiner's user avatar
  • 3,079