Barrycentric coordinates for a polytope

[salman7866]

For the past few weeks, I have been searching about this topic: Suppose we are given a convex polytope having vertices say \begin{equation} A_1, A_2,...,A_n \end{equation} where each \begin{equation}A_i, i=1,...,n \end{equation} represent a matrix - In fact its a convex polytope whose vertices are matrices. How can we verify if a given a matrix say \begin{equation} A_t \end{equation} can be written as a convex combination of the vertices.

To elaborate more: I am constructing this polytope to encompass a time-varying matrix; say denoted by A(t) where the variable "t" is varying between some upper and lower bounds t_{min} and t_{max}. Using these bounds of t, I find vertices for a polytope denoted by \begin{equation}A_1,...,A_n \end{equation}. Now how do I prove that given A(t) where \begin{equation}t_min \leq t \leq t_max \end{equation} can be written as convex combination of the vertices \begin{equation}A_1,...,A_n \end{equation}. Any help would be appreciated.

 

[jvicens]

Hello there,

Thank you for sharing your question and interest in this topic. I would like to provide some insight and suggestions on how you can approach this problem.

First, it is important to understand the concept of convex combinations. A convex combination is a linear combination of points where the coefficients are non-negative and sum up to 1. In other words, it is a weighted average of points within a convex set.

Now, let's consider your scenario where you have a time-varying matrix A(t) and a set of vertices A_1, A_2,...,A_n. In order to prove that A(t) can be written as a convex combination of the vertices, you need to show that there exist non-negative coefficients c_1, c_2,...,c_n such that c_1A_1 + c_2A_2 +...+ c_nA_n = A(t) and c_1 + c_2 +...+ c_n = 1.

One approach to verify this is to use the concept of linear independence. If the vertices A_1, A_2,...,A_n are linearly independent, then there exists a unique solution for the coefficients c_1, c_2,...,c_n. In this case, you can solve for the coefficients using techniques such as Gaussian elimination or matrix inversion.

However, if the vertices are not linearly independent, then there may be multiple solutions for the coefficients. In this case, you can use optimization techniques to find the coefficients that minimize a certain objective function, such as the sum of squared errors between A(t) and the convex combination of the vertices.

Another approach is to use the concept of extreme points. In a convex set, extreme points are the points that cannot be expressed as a convex combination of other points in the set. If your vertices A_1, A_2,...,A_n are extreme points, then A(t) can be written as a convex combination of the vertices.

In summary, to prove that A(t) can be written as a convex combination of the vertices A_1, A_2,...,A_n, you can use techniques such as linear independence, optimization, or extreme points. I hope this helps and good luck with your research!