What is a Convex Hull and How Does it Relate to Positive Definite Objects?

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In summary, a convex hull is the smallest convex set that contains a given set of points. It can be visualized by experimenting with different numbers of points on a sheet of paper, where 2 points make a line segment, 3 points make a triangle, and 4 points make a tetrahedron. Convex sets have the same dimension as the space they sit in, which is why a circle or square are not considered convex hulls. The definition of a convex set involves it being a positive definite object, but the exact definition is uncertain.
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Tenshou
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I don't understand what a convex hull is and what it does. does anyone think they can explain what exactly it is?
 
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The convex hull of A is the smallest convex set that contains A.

If you want to see what it is, then you should experiment. Draw 2 points on a sheet of paper. What do you think that the convex hull is?
What about 3 points?
What about 4 points?
 
  • #3
Thank you micromass, I think I get it... I think I understand that the two points should be a line segment, 3 points a triangle and 4 a tetrahedron. I just don't get why those are convex sets, is it because the dimension of the line segment(or the other stuff) is equal to the dimension of the plane that it sits in? Why isn't a circle a convex hull or a square? are they convex sets?
 
  • #4
Tenshou said:
Thank you micromass, I think I get it... I think I understand that the two points should be a line segment, 3 points a triangle and 4 a tetrahedron. I just don't get why those are convex sets, is it because the dimension of the line segment(or the other stuff) is equal to the dimension of the plane that it sits in? Why isn't a circle a convex hull or a square? are they convex sets?

what is the definition od a convex set?
 
  • #5
I do not know, but I think it talks about some set which is Positive Definite "object", but I am not sure :/ I mean just thinking about convex things that is what I think it could be.
 

FAQ: What is a Convex Hull and How Does it Relate to Positive Definite Objects?

What is a convex hull?

A convex hull is a geometric shape that contains all points within a set, while also being the smallest possible shape to do so. It can be thought of as the outer boundary or envelope of a set of points.

How is a convex hull calculated?

The most common way to calculate a convex hull is by using the "gift wrapping" algorithm, also known as the Jarvis march. This algorithm starts at the leftmost point and wraps an imaginary string around the points, creating the convex hull. Other algorithms, such as Graham's scan and Quickhull, can also be used.

What are the applications of convex hulls?

Convex hulls have various applications in different fields such as computer graphics, image processing, robotics, and operations research. They are used to determine the boundary of an area or shape and can also be used for collision detection in computer simulations.

What is the difference between a convex hull and a concave hull?

A convex hull is a shape that contains all points within a set, while a concave hull is a shape that contains the points and also has indentations or "dents". In other words, a concave hull is not a continuously smooth boundary like a convex hull.

Can a convex hull be applied to non-geometric data?

Yes, a convex hull can be applied to any set of points, regardless of whether they represent geometric or non-geometric data. However, the shape may not have a clear interpretation in non-geometric cases.

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