23
votes

Accepted

### Can every simple polytope be inscribed in a sphere?

Not all simple polytopes are incribable, e.g. the dual of the cyclic polytope $C_4(8)$ is simple and not inscribable, as shown recently in Combinatorial Inscribability Obstructions for Higher-...

22
votes

Accepted

### Why are we interested in permutahedra, associahedra, cyclohedra, ...?

Philosophical questions deserve philosophical answers, so I am afraid no amount of references and specific results will probably satisfy you. Let me try to explain it in a somewhat generic way.
...

17
votes

### Why are we interested in permutahedra, associahedra, cyclohedra, ...?

The origins of associahedra go back to the thesis work in homotopy theory of Jim Stasheff in the early 1960's.
He did graduate work at Oxford, working with Ioan James, who in the mid 1950's had ...

13
votes

### Convex lattice polygons with equal area and perimeter

Pick's theorem says these two convex lattice polygons have area
$$i+\frac{b}{2}-1 = 4 + 10/2 -1 = 8 \;,$$
and they both have perimeter $8 + 2 \sqrt{2}$.
You can see I've "bumped out" two ...

12
votes

### Why are we interested in permutahedra, associahedra, cyclohedra, ...?

In my opinion there are two answers to this question.
The first is that these particular classes of polytopes have fascinating combinatorial properties and structure. Presumably you're aware of the ...

11
votes

### Why are we interested in permutahedra, associahedra, cyclohedra, ...?

There are remarkable combinatorial formulas for the face numbers and the volumes (of certain geometric realizations of) of these polytopes and a more general family ("generalized permutohedra" a.k.a. "...

9
votes

Accepted

### Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?

The answer is No, there are no other such polytopes. The proof is quite laborious in parts, and I wrote it down in this article.
Theorem. In dimension $d\ge 4$, an edge-transitive polytope is vertex-...

9
votes

### Why are we interested in permutahedra, associahedra, cyclohedra, ...?

I'm not an expert, but perhaps this could help:
Hohlweg, Christophe. "Permutahedra and associahedra: Generalized associahedra from the geometry of finite reflection groups." arXiv:1112.3255 (2011): ...

8
votes

Accepted

### Is there a 4-polytope without 3-gonal and 4-gonal faces, other than the 120-cell?

There are other polytopes. To construct one let's do the following. Remember first that in the hyperbolic $4$-space there exists a regular compact right-angled 120-cell. Here, right-angled means that ...

8
votes

Accepted

### Is a polytope that has in-spheres for faces of all dimensions already regular?

This is true in all dimensions, and can be proved by induction (on $d$) applied to the following (slightly stronger) hypothesis:
Theorem: If $P$ is a convex $d$-polytope with $k$-in-spheres for all $...

7
votes

Accepted

### Realizing spherical complexes as convex polytope

Another way to phrase your question is "whether every complete fan is (combinatorially equivalent to) the face fan (or the normal fan) of a convex polytopes". The answer is No in dimension $...

6
votes

Accepted

### Triangulation of a simplex

You are looking for the edgewise subdivision:
Edelsbrunner, H.; Grayson, D. R., Edgewise subdivision of a simplex, Discrete Comput. Geom. 24, No. 4, 707-719 (2000). ZBL0968.51016.
The basic idea is to ...

6
votes

### A rational polytope that is not a 01-polytope?

Rather obvious in retrospect, but all 01-polytopes are inscribed (the vertices lie on a sphere). So if $P$ is non-inscribable, then it can't be a 01-polytope. There are non-inscribable rational ...

5
votes

### What does it mean polynomials share Newton polytope?

Newton polytopes and the polynomials they support
We will use the standard notion $\mathbf{x}^{\mathbf{a}} := \prod_{i=1}^{n} x_i^{a_i}$ to represent monomials in a multivariate (Laurent) polynomial ...

5
votes

### “Totally transitive” polytopes which are not regular

It is possible to have an abstract polytope where the automorphism group acts transitively on the faces of each rank (fully transitive) but does not act transitively on flags (not regular).
For any ...

5
votes

### Approximation of convex hull in high dimension

This recent paper
Sartipizadeh, Hossein, and Tyrone L. Vincent. "Computing the Approximate Convex Hull in High Dimensions." arXiv:1603.04422 (2016).
includes a summary of previous work on ...

5
votes

Accepted

### Edges of the contact polytope of the Leech lattice

Using the unimodular scaling of the Leech lattice, the length of each minimal vector is $\sqrt{4}$. Fixing a particular minimal vector $u$, the remaining minimal vectors $v$ are:
1 vector $v$ with $\...

5
votes

### 4-polytopes with only one kind of regular facet

This is Theorem 1 (actually, Satz 1) of Roswitha Blind, Konvexe Polytope mit kongruenten regulären $(n- 1)$-Seiten im $\Bbb{R}^n$ $(n \ge 4)$, Comment. Math. Helvetici 54 (1979) 304--308. The short ...

5
votes

### Solid angles at points in an orthosimplex

Note that $x+\varepsilon y\in K$ for small $\varepsilon$ if and only if $y$ satisfies the inequalities $y_i\leqslant y_{i+1}$ if $x_i=x_{i+1}$; $y_1\geqslant 0$ if $x_1=0$; $y_n\leqslant 0$ if $x_n=0$....

5
votes

Accepted

### Convex lattice polygons with equal area and perimeter

This answers question 2 as well. But I think both questions are way more suitable for math.stackexchange.
I add another example because, unlike the first, it has the property that multiplied by $I^{n-...

4
votes

Accepted

### Are there any more polytopes whose 2-faces are identical 4-gons?

In fact, there are many to be found on Wikipedia under isogonal figures, even in three dimensions.
Examples in dimension four are obtained as dual polytope of runcinated 4-simplex or runcinated 24-...

4
votes

### Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?

If you consider a tiling of 3-space to be a 4-dimensional polytope, then the Rhombic dodecahedral honeycomb would work.
Other possibilities are limited by the potential 3-faces.
Because every edge ...

4
votes

### Derive a vertex representation of a permutohedron from its linear-inequalities form

Observe that your set of inequalities is $S_n$-invariant, hence your polytope is, hence your set of vertices is. So it's enough to understand the case $x_1 \leq \ldots \leq x_n$.
Now you don't need ...

4
votes

### Is a polytope that has in-spheres for faces of all dimensions already regular?

In $R^3$, since the spheres are concentric, not only all faces are regular, but also all edges are of the same length, and all faces are inscribed in circles of the same radius, hence are congruent. ...

4
votes

Accepted

### A polytope with congruent facets and an insphere that is not facet-transitive?

First, a simple remark: If a polytope with congruent facets is inscribed in a sphere, then it is circumscribed about a sphere as well, and the two spheres are concentric.
Next, there is a series of ...

4
votes

### Do Bernoulli polynomials know about face vectors?

Following up on i9Fn's suggestion, there does seem to be at least one more case along these lines: $$660F_{11}(n) = 60n^{11} + 330n^{10} + 550n^9 - 660n^7 + 660n^5 - 330n^3 + 50n.$$
This might be ...

4
votes

### What are the "simplest" polytopes with an automorphism group of $\mathrm M_{11} \hspace {-1.25pt} $?

$\color {red} {\textbf {WARNING}}$
I believe @M.Winter has shown in the comments that the following answer is, in fact, $\color {red} {\textbf {incorrect}}$. It would appear that I have merely ...

4
votes

Accepted

### Quasiconformal map from a subset of $\mathbb{C}$ to a polytope

Yes, there exists a quasiconformal map (even a homeomorphism) from the Riemann sphere
to a polytope. Embedding to $R^3$ is irrelevant here, all we need is the intrinsic
metric, which is a flat metric ...

4
votes

### Can a dodecahedron be deformed into a great stellated dodecahedron?

Talking with Saul Schleimer, we came up with the following:
Orthogonally project the great stellated dodecahedron into the $z=0$ plane, choosing a direction that does not result in any zero length ...

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