A linear programming, polyhedron and extreme points

  • A
  • Thread starter Thread starter mertcan
  • Start date Start date
mertcan
Messages
343
Reaction score
6
Hi,

First assume that there is a polyhedron P, where Ax<=b and x is free variable whose dimension is n. Besides, rank(A) = n. I really wonder if extreme points in a polyhedron can be linearly dependent? I used even ChatGPT, but it includes some shaky calculations while proofing. In short, if extreme points in a polyhedron can be linearly dependent or not, could you provide me with a proof?

Thanks
 
Physics news on Phys.org
mertcan said:
Hi,

First assume that there is a polyhedron P, where Ax<=b and x is free variable whose dimension is n. Besides, rank(A) = n. I really wonder if extreme points in a polyhedron can be linearly dependent? I used even ChatGPT, but it includes some shaky calculations while proofing. In short, if extreme points in a polyhedron can be linearly dependent or not, could you provide me with a proof?

Thanks
As an example you can consider a triangle in the plane R^2 with vertices
A=(0,1)
B=(1,0)
C=(1,1)

x<=1
y<=1
x+y>=1 ( -x-y<=-1 )
 
Last edited:
very thanks for your reply. Well, may I ask if ANY TWO extreme points in a polyhedron can be linearly dependent or not (those 2 extreme points are multiple of each other or not), could you provide me with a proof?

Thanks
 
It's trivial that you can get two points to be linearly dependent. For example the square with vertices (0,0), (0,1), (1,0) and (1,1). (0,0) Is linearly dependent with every other vertex.
 
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
Back
Top