# Variational methods - properties of convex hull

For (d), use the fact that Co(A) is a subset of Co(B) when A is a subset of B.In summary, the properties of convex hull are as follows: (a) The convex hull of the convex hull of a set A is equal to the convex hull of A. (b) The convex hull of the union of two sets A and B is a superset of the union of the convex hulls of A and B. (c) If a set A is a subset of a set B, then the convex hull of the union of A and B is equal to the convex hull of B. (d) If a set A is a subset of a set B, then the convex hull of A
Show the following properties of convex hull:
(a) Co(CoA) = Co(A)
(b) Co(AUB) $$\supseteq$$Co(A) U Co(B)
(c) If A$$\subseteq$$B then Co(AUB)=Co(B)
(d) If A$$\subseteq$$B then Co(A)$$\subseteq$$Co(B)

The definition of a convex hull is a set of points A is the minimum convex set containing A.
(c) is quite trivial and i can get it.
but i am wondering about (a) and (b) and (d), anyone know if (d) is proven using (b) and (c) or is there another method of doing it.
I am having difficulty explaining (a), I think i understand why they are equal.. it is quite obvious, but i can't explain it well.
and as for (b) i am also lost for words for the explanation

any help would be greatly appreciated

How about this explanation for (a): The minimum convex set of a convex set is itself and the result follows. For (b), pick two points in the union of Co(A) and Co(B) and show that they're also in Co(A U B).

## 1. What is the definition of a convex hull?

A convex hull is the smallest convex set that contains all the points in a given set of points. It can be thought of as the "envelope" that wraps around the points.

## 2. What are the key properties of a convex hull?

The key properties of a convex hull are: it is convex, it contains all the points in the original set, and it has the smallest possible area or volume compared to other enclosing shapes.

## 3. How is the convex hull related to variational methods?

In variational methods, the convex hull is used as a tool to find optimal solutions to problems. It allows for the reduction of a complex problem to a simpler one, making it easier to find a solution.

## 4. Can the convex hull be computed efficiently?

Yes, there are efficient algorithms for computing the convex hull. One popular algorithm is the Graham scan, which has a time complexity of O(n log n).

## 5. How is the convex hull used in real-world applications?

The convex hull has various applications in fields such as computer graphics, image processing, and data analysis. It is used for tasks such as shape recognition, data clustering, and finding the "best fit" solution to a problem.

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