2
votes
0answers
50 views

convexity in oriented matroid theory

I would like to try to solve the following problems. If someone knows how to prove at least part (a), could you show me the proof? I am having a LOT of trouble understanding oriented matroid theory. ...
2
votes
1answer
186 views

Von Neumann's minimax theroem and Carathéodory's theorem

In J.F. Mertens(1986)(Paywall), there's a neat proof of a version of Von Neumann's minimax theroem. But I can't understand how Carathéodory's theorem is invoked. Suppose, in a two-person zero sum ...
10
votes
1answer
133 views

How many hyperplanes does it take to separate $n$ points in $\mathbb{R}^m$?

By 'separate', I mean that each point lies in its own little region/cell. For instance, it takes a minimum of $P = 4$ lines to separate $n = 7$ points in $\mathbb{R}^2$ ($m=2$), assuming that no 3 ...
2
votes
0answers
239 views

Find a tight upper bound of the following expectation.

I am stuck in finding a tight upper bound (as tight as possible) of the following expectation $$E\left [ (1-a\cdot b^{X})^{m} \right ]$$ where $X\sim B(n-1,p)$ is a binomial random variable.In ...
1
vote
0answers
175 views

how to solve the following expectation? closed-form expression or approximation

Suppose there is a binomial random variable $X\sim B(n-1,p)$,how to solve the following expectation $$E[(1- b^{X})^{m}]$$ where $b\in (0,1]$ and $m\in \mathbb{N} $ are all constants.I have tried my ...
1
vote
1answer
129 views

Submodularity of the product of two non-negative, monotone increasing submodular functions

I'm trying to prove the submodularity of the product of two non-negative, monotone increasing submodular functions Formally, we have $f$ and $g$ are submodular functions, that is, ...
12
votes
1answer
323 views

What is the probability of having a pentagon in 6 points

If the probability that $5$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$ occur to be the vertices of a ...
15
votes
1answer
282 views

Expected size of subset forming convex polygon.

If there are $4$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$, what is the expected largest size (or ...
1
vote
1answer
82 views

Sub-Modular Set Function?

For a fixed set $B$ and for sets $A_i ,\forall i \in \left \{ 1,2,\dots,n \right \}$ , I define $f(A_i)=\frac{|A_i \cap B|}{2|A_i|-|A_i\cap B|}$, where $|A_i|>0$ is the cardinality of set $A_i$. ...
0
votes
1answer
51 views

If E is a subset of a lattice closed under addition then is the intersection of E with the opposite of some translate finite?

this seems intuitive to me but I'm struggling to prove it (is it false?). Let $E$ be a subset of a lattice (free abelian group of finite rank) closed under addition, containing the origin and such ...
2
votes
1answer
283 views

Applications of Helly's theorem to problem solving

Helly's Theorem states the following: Suppose that $X_1,X_2,...,X_n$ are convex sets in $\mathbb{R}^d$, such that for any $|I|\leq d+1$, $\cap_{i\in I}X_i \neq \emptyset$. Then $\cap_{i=1}^{n}X_i ...