# Tagged Questions

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### Status of Barany's conjecture?

One of Barany's most intriguing conjectures is about the $f$-vectors of convex polytopes. It asks: Let $P$ be a convex $d$-polytope. Is it always true that $f_k \geq \min(f_0, f_{d-1})$? A ...
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### Triangulations of special polyhedra

Let $A_1,A_2,A_3 \in \mathbb{N}^3$ be three points in space all lying in some plane $x+y+z=d$ where $d$ is a positive integer. If $\{e_1,e_2,e_3\}$ is the standard basis in $\mathbb{R}^3$, we can ...
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### Computational complexity of deciding isomorphism of rational polyhedral cones

Let $C,C'$ be rational polyhedral cones in $\mathbb R^n$ both with non-empty interior. Rational means they are generated by vectors with rational entries. One says that $C,C'$ are isomorphic if there ...
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### References to proofs of a theorem by Van Kampen-Flores

Theorem (Van Kampen-Flores 1930s) From any 7 points in four-dimensional space one can choose two disjoint triples such that the triangles with vertices at the triples intersect each other. This ...
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### Maximizing ratio volume/diameter^n by an affinity

Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$. It is clear that there is a constant ...
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### Volume-like property to upper bound lattice points in a convex body

The following question arises in passing in a joint paper that I am working on. Let $K$ be a centrally symmetric convex body in an $n$-dimensional real vector space $V$ which contains a lattice $L$. ...
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### On the number of lines of given points

Hi all, I have a question Concerning Beck's theorem. I have read it from http://en.wikipedia.org/wiki/Beck%27s_theorem and I have two questions : I suppose Beck's theorem doesn't hold when instead ...
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### Realisation of convex polygons with an interior point from combinatorial data

A convex polygon $P$ having an interior point $A$ in generic position (not on any line defined by two vertices) can be encoded combinatorially by associating to every vertex $v$ of $P$ the unique edge ...
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### Placing points on a sphere so that no 3 lie close to the same plane

Motivation I am working with arbitrary parallelopiped tilings given by projection from a higher dimensional space. The collection of tiles, and some properties of the higher dimensional space are ...
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### Applications of ham sandwich type results. References? A general principle?

Lately there has been a lot of interest on applications of the ham sandwich theorem and related results. There is a bunch of lecture notes and surveys that touch upon the subject. I dont know of any ...
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### Extreme points of transportation polytope

I'm interested in $n \times m$ joint probability tables with prescribed row and column marginals. Such tables form a convex set known as the transportation polytope. What are the extreme points of ...
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### Maximum of a function on $d-$dimensional convex compact sets

Let $\mathcal C_d$ denote the set of all $d-$dimensional convex compact subsets with barycenter at the origin of the $d$-dimensional Euclidean space $\mathbb E^d$. Given an element $C\in\mathcal C_d$ ...
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### The number of hyperplanes determining the integer points of a polyhedron

This question is inspired by this one. Let $P \subset \mathbb R_{+}^n$ be a convex polyhedron whose complement in $\mathbb R_{+}^n$ has finite volume. Let $Int(P) = P \cap \mathbb N^n$. (For ...