# Finding the convex hull of m x n matrices

Given a set of $$\mathbb{R}^{m \times n}$$ matrices, I would like to find the matrices forming the vertices of their convex hull.

Would this be the same problem as finding the convex hull of a set of vectors in $$\mathbb{R}^{m + n}$$? ie: Could I simply reshape the matrices into vectors and proceed with known algorithms to find the convex hull, and then convert back into $$\mathbb{R}^{m \times n}$$?

PS: Please be kind. I am a engineer with limited experience in this field. If you could point me to any references that could be helpful, I's appreciate it.

Thanks

Edit:

1. My set of $$\mathbb{R}^{m \times n}$$ matrices is finite and each matrix is discrete.

2. Some additional clarification: I am building this set up one matrix at a time. Each time I have a new matrix, I want to test if the new matrix is contained within the convex hull formed by the matrices already in the set. If it is, then I can ignore it. If it is not in the convex hull, then I add it to the set and extend the convex hull to include the new matrix.

• Even in $\mathbb{R}^2$, the problem seems unclear. Circle is a convex shape. How do you define a vertex for a convex shape like circle? Also, on second paragraph of your question, you talk about finding the convex hull. Do you want to find the vertices given a set or find the convex hull given the vertices? – stressed out Feb 27 '19 at 0:38
• This is a hard problem in general even with set of matrices with harmless looking properties. – chhro Feb 27 '19 at 0:39
• stressed out, my set includes a finite number of matricies, so your example of a circle is not relevant to my problem, but is is a good example which illustrates why this a hard problem in general. Regarding the second paragraph, maybe a quick example will help explain. If I have a set of, say, 10 matrices, and 5 of these form the vertices of the convex hull which contains all 10, I want an algorithm that identifies these 5 vertex matrices. – drambugh Feb 27 '19 at 5:13

Since $$\mathbb R^{m\times n}$$ is isomorphic as a vector space to $$\mathbb R^{mn}$$ (note that that‘s not a plus) via the obvious map (taking all entries), you can indeed do the computation in $$\mathbb R^{mn}$$.