# Tagged Questions

**4**

votes

**1**answer

91 views

### The number of facets of a polyhedron under linear transformation

Consider a (not necessarily bounded) convex polyhedron $P\subset \mathbb{R}^n$ which has $k$ facets.
Let $L:\mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation.
Question1: Is there a fixed ...

**5**

votes

**1**answer

265 views

### Linear transformation of a polyhedron

Is there a simple proof that shows:
Linear transformation of a $\mathcal{H}$-polyhedron (i.e. the intersection of
finitely many closed half-spaces) is a $\mathcal{H}$-polyhedron.
Minkowski sum of ...

**3**

votes

**2**answers

121 views

### 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 ...

**4**

votes

**2**answers

108 views

### The maximal discrete parallelepiped in a convex body

Does the positive constant $c_d$, depending only from dimension, with the following property exist?
Property:
for every convex body $K\subset \mathbb R^d$ there exists parallelepiped $P\subset K$ ...

**4**

votes

**1**answer

192 views

### n-simplex in an intersection of n balls

Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...

**3**

votes

**1**answer

102 views

### Covering a convex body with its smaller homothetic copies

Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C + x = \{ \lambda c + x \mid c\in C \}$ for some $x\in R^d$ is called a homothetic copy of $C$. The ...

**1**

vote

**1**answer

148 views

### Helly's number from biconvex functions

Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...

**17**

votes

**3**answers

1k views

### Research trends in geometry of numbers?

Geometry of numbers was initiated by Hermann Minkowski roughly a hundred years ago. At its heart is the relation between lattices (the group, not the poset) and convex bodies. One of its fundamental ...

**11**

votes

**1**answer

320 views

### Detecting a hidden convex body with line probes

Imagine that, somewhere inside an origin-centered, unit-radius sphere
$S$ in $\mathbb{R}^3$,
sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$,
with $\alpha < 1$ the fraction of ...

**0**

votes

**1**answer

338 views

### 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 ...

**7**

votes

**3**answers

250 views

### Expected minimum face angle of random convex polyhedron in $\mathbb{R}^3$

Let $P_n$ be a "random convex polyhedron" in $\mathbb{R}^3$ of $n$ vertices, where "random" could follow any one
of a number of models:
(1) the convex hull of $n$ points randomly and uniformly ...

**4**

votes

**4**answers

877 views

### Delaunay triangulations and convex hulls

This is a reference request.
I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you ...

**17**

votes

**2**answers

779 views

### 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 ...

**1**

vote

**1**answer

612 views

### 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 ...

**2**

votes

**3**answers

817 views

### Draw a Random Line Through a Voronoi Tessellation, What is the Average Number of Voronoi Cell the Line Intersects?

Update: problem reformulation
Following the advice in comments, I now restate my problem using Voronoi
tessellation.
Given a unit hypercube $H_n=\{(x_1,\ldots,x_n)\in \mathbb{R}^n: 0\leq x_i\leq
...

**10**

votes

**2**answers

569 views

### Helly theorem + Nerve

Consider nerve $\mathcal N$ of a finite set of convex sets in $\mathbb R^n$.
Helly theorem says that $\mathcal N$ is completely determined by its $n$-skeleton, say $\mathcal N_n$.
It seems that not ...

**1**

vote

**2**answers

596 views

### Is there always a parallelogram cross-section of parallelepiped contained in the smallest box

Let $M$ be a centered parallelepiped, the intersection of $M$ and any plane $P$ that passes through the origin is a parallelogram or hexagon. Each parallelogram or hexagon has a cubic box that is the ...

**3**

votes

**0**answers

123 views

### Dimension of convex arrangements for hypergraphs

Suppose you have a hypergraph H on n vertices. Let d be the smallest integer such that we can find an arrangement A of convex subsets in Rd so that H represent the intersections of sets in A.
Has ...

**16**

votes

**3**answers

1k views

### Cutting convex sets

Any bounded convex set of the Euclidean plane can be cut into two convex pieces of equal area and circumference.
Can one cut every bounded convex set of the Euclidean plane into an arbitrary number ...