# Category Archives: Convexity

# Test Your Intuition (2)

**Question:** Let be the cube in centered at the origin and having -dimensional volume equal to one. What is the maximum -dimensional volume of when is a hyperplane?

Can you guess the behavior of when ? Can you guess the plane which maximizes the area of intersection for ?

Test your intuition **before** reading the rest of the entry.

# Sarkaria’s Proof of Tverberg’s Theorem 2

Karanbir Sarkaria

## 4. Sarkaria’s proof:

**Tverberg’s theorem (1965):** Let be points in , . Then there is a partition of such that .

**Proof:** We can assume that . First suppose that the points belong to the -dimensional affine space in of points with the property that the sum of all coordinates is 1. Next consider another -dimensional vector space and vectors in , such that is the only linear relation among the s. (So we can take as the standard basis in and . )

Now we consider the tensor product .

Nothing to be scared of: can be regarded just as the -dimensional vector space of *d+1* by matrices. We can define the tensor product of two vectors and , as the (rank-one) matrix .

Consider in U the following sets of points:

…

…

Note that 0 is in the convex hull of every . Indeed it is the sum of the elements in . (And thus . )

By the Colorful Caratheodory Theorem we can Continue reading

# Sarkaria’s Proof of Tverberg’s Theorem 1

Helge Tverberg

Ladies and gentlemen, this is an excellent time to tell you about the beautiful theorem of Tverberg and the startling proof of Sarkaria to Tverberg’s theorem (two parts). A good place to start is Radon’s theorem.

## 1. The theorems of Radon, Helly, and Caratheodory.

### Radon’s theorem

**Radon’s theorem: **Let be points in , . Then there is a partition of such that . (Such a partition is called a **Radon partition**.)

**Proof: **Since the points are affinely dependent. Namely, there are coefficients, not all zero, such that and . Now we can write and , and note that

(*) ,

and also . This last sum is positive because not all the s are equal to zero. We call it .

To exhibit a convex combination of which is equal to a convex combination in just divide relation (*) by . **Walla.**

This trick of basing a partition on the signs of the coefficients repeats in other proofs. Take note!

Radon used his theorem to prove Helly’s theorem.

### Helly’s theorem

**Helly’s theorem**: For a family , , of convex sets in , if every of the sets have a point in common then all of the sets have a point in common.

**Proof:** It is enough to show that when if every of the sets have a point in common then there is a point in common to them all. Let . In words, is a point that belongs to all the s except perhaps to . We assumed that such a point exists for every .

Now we can apply Radon’s theorem: Consider the Radon partition of the points, namely a partition of such that . Let . Since belongs to the convex hull of and every for belongs to every for every not in we obtain that belongs to every for not in . By the same argument belongs to every for not in . Since and are disjoint, the point belongs to all s. **Ahla.**

The proof of Helly’s theorem from Radon’s theorem as described on the cover of a book by Steven R. Lay

### Caratheodory’s theorem

**Caratheodory’s theorem: **For , if then for some , .

Like for Radon’s theorem, there is a simple linear algebra proof. We can assume that is finite; we start with a presentation of as a convex combination of vectors in , and if we can use an affine dependency among the vectors in to change the convex combination, forcing one coefficient to vanish.

Without further ado here is Tverberg’s theorem.

## 2. Tverberg’s Theorem

**Tverberg Theorem (1965):**Let be points in , . Then there is a partition of such that .

So Tverberg’s theorem is very similar to Radon’s theorem except that we want more than two parts! Continue reading

# Helly, Cayley, Hypertrees, and Weighted Enumeration III

This is the third and last part of the journey from a Helly type conjecture of Katchalski and Perles to a Cayley’s type formula for “hypertrees”. (On second thought I decided to divide it into two devoting the second to probabilistic questions.) This part will include several diversions, open problems, and speculations.

### 11. How to make it work – the matrix tree theorem

Our high dimensional extension to Cayley’s theorem reads:

where the sum is over all d-dimensional simplicial complexes K on n labelled vertices, with a complete (d-1)-dimensional skeleton, and which are Q-acyclic, namely all their (reduced) homology groups with rational coefficients vanish.

Looking at the various proofs of Cayley’s formula (there are many many many beautiful proofs), the proof that I know to apply is the one based on the matrix tree theorem.

Consider the signed incidence matrix A’ between all (d+1)-subsets and all d-subsets of {1,2,…,n} that represents the boundary operator of simplicial homology. The rank of this matrix is , and just like in the ordinary matrix tree theorem you delete rows to be left with linearly independent rows. Here you delete all rows corresponding to sets containing ‘n’ and you are left with a matrix A. Now we compute the determinant of directly, and compare the result to a computation based on the Cauchy-Binet Formula.

The eigenvalues of are the eigenvalues of the d-th Laplacian of the complete d-dimensional simplicial complex with n vertices. It is easy to inspect what they are and the determinant of is indeed .

The many square determinants correspond to d-dimensional simplicial complexes K on our labelled set of vertices, which satisfy and . Now if K has non-vanishing d-th homology, the determinant is zero. If K is a Q-acyclic simplicial complex (i.e., its (reduced) homology groups with rational coefficients are trivial) then it has a non zero determinant. So far, it is like trees, but next comes a surprise. The contribution of K is the square of the number of elements in , the (d-1)th homology group of K. This torsion group is finite, but for d >1 it need not be trivial.

Emily Peters presents the matrix-tree theorem. From “The Sarong Theorem Archive” – an electronic archive of images of people proving theorems while wearing sarongs.

### 12. An even simpler use of Cauchy-Binet worth knowing

Consider the matrix A, whose columns are all +1 -1 vectors of length n. Computing , via Cauchy-Binet Formula (or by other easy methods) asserts that the expected value for all +1/-1 matrices behaves roughly like . This was observed by Turan and Szekeres who also found a formula for the sum of the fourth powers of all 0-1 n by n matrices. See a leter paper by Turan. (I am not aware of a similar formula for the fourth power of the size of the homology groups for hypertrees.) Much is known about the determinant and related properties of random 0-1 matrices and the analogy between torsion in the homology groups of random complexes and determinants of random matrices looks like a good analogy.

### 13. Torsion

One consequence of the formula compared to the total number of available simplicial complexes is that the torsion group is typically huge. (For d>1, the expected value of for Q-acyclic complexes counted by the formula, is asymptotically larger than ; I am not aware of explicit examples with such a huge torsion group.)

How should we think about torsion in homology? It seems that thinking about the size of the torsion as a behaving like the determinant of a random matrix, may give a good intuition for many cases.

### 14. Extending other proofs for Cayley’s theorem?

Cayley’s counting trees theorem has many wonderful proofs. Can any other proof extend to the case of Q-acyclic simplicial complexes? For example, one proof relies on the exponential theorem that relates the exponential generating functions for connected and general graphs with a certain property P. (Followed by the Lagrange inversion formula.) Is there an analog of the exponential formula when connectivity is replaced by higher homology? Is there any analog of Prüfer sequences? I am not aware of any other proof that works.

### 15. Weights to the rescue of other conjectures?

Can we use subtle weights to save other promising but false enumerative conjectures? The farthest reaching fantasy in this direction Continue reading

# Helly’s Theorem, “Hypertrees”, and Strange Enumeration II: The Formula

In the first part of this post we discussed an appealing conjecture regaring an extension of Cayley’s counting trees formula. The number of d-dimensional “hypertrees” should somehow add up to . But it was not clear to us which complexes we want to count and how. This counting problem started from a Helly type conjecture proposed by Katchalski and Perles.

For d=2 n=6 the situation was confusing. We had 46608 complexes that were collapsible. Namely, for these complexes it is possible to delete all triangles one at a time by removing in each step a triangle T and an edge E which is contained only in T. Once all triangles are removed we are left with a spanning tree on our 6 vertices. (Five out of the 15 edges survive). In addition, there were 12 simplicial complexes representing 6-vertex triangulations of the real projective plane.

We will continue the discussion in this part, show how the conjecture can be saved and at what cost. We will also discuss the solution of the Perles-Katchalski conjecture – a Helly’s type conjecture that we started with. In the third part we will explain the proof and mention further related results and problems, discuss higher Laplacians and their spectrum, and mention a few related probabilistic problems.

### **We ended part one with the question “What can we do?”**

### 8. How to make the conjecture work

With such a nice conjecture we should not take **no** for an answer. To make the conjecture work we need to count each of the twelve 6-vertex triangulations of the real projective plane, four times. Four is the square of the number of elements in . This is the difference in higher dimensions, a Q-acyclic complex need not be Z-acyclic. Homology groups can have non trivial **torsion**. In our case can be a non trivial finite group.

Here is the theorem:

where the sum is over all d-dimensional simplicial complexes K on n labelled vertices, with a complete (d-1)-dimensional skeleton, and which are Q-acyclic, namely all their (reduced) homology groups with rational coefficients vanish.

Looking at the various proofs of Cayley’s formula (there are many many many beautiful proofs and more), which one (or more) would you expect to extend to the high dimensional case? We will answer this question in part III. Can you guess? Continue reading

# Helly’s Theorem, “Hypertrees”, and Strange Enumeration I

### 1. Helly’s theorem and Cayley’s formula

**Helly’s theorem** asserts: For a family of n convex sets in , n > d, if every d+1 sets in the family have a point in common then all members in the family have a point in common.

**Cayley’s formula **asserts: The number of trees on n labelled vertices is .

In this post (in two parts) we will see how an extension of Helly’s theorem has led to high dimensional analogs of Cayley’s theorem.

left: Helly’s theorem demonstrated in the Stanford Encyclopedia of Philosophy (!), right: a tree

### 2. Background

This post is based on my lecture at Marburg. The conference there was a celebration of new doctoral theses Continue reading