Convexity of set A = {(x,y) in R^2 | x^4+y^4 =< 1, x>=0

peteryellow · 2008-03-18T14:34:39+00:00

How can I show that the set A = {(x,y) in R^2 | x^4+y^4 = =0 y>=0} is convex.

Thread starter peteryellow
Start date Mar 18, 2008
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In summary, convexity is a property of a set where any line segment connecting two points in the set lies entirely within the set. The set A is convex as observed by graphing it. Other ways to determine if a set is convex include using the definition of convexity and observing if the intersection of convex sets is also convex. Convexity plays a crucial role in optimization problems and is used in various fields such as geometry, economics, and computer science.

Mar 18, 2008

peteryellow

How can I show that the set A = {(x,y) in R^2 | x^4+y^4 =< 1, x>=0 y>=0} is convex.

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Mar 21, 2008

CHatUPenn

I am sure it can be shown by definition, but I propose an easy way (not rigorous though)

The function is (weakly) convex
The lower contour set (=<1) of a convect function is convex.

1. What is the definition of convexity?

Convexity refers to the property of a set where any line segment connecting two points in the set lies entirely within the set. In other words, a convex set does not contain any indentations or "dents" that would cause a line segment to cross the boundary of the set.

2. Is the set A convex?

Yes, the set A is convex. This can be observed by graphing the set and seeing that any line segment connecting two points in the set lies within the set.

3. How can I determine if a set is convex?

One way to determine if a set is convex is by graphing the set and observing if any line segment connecting two points in the set lies within the set. Another way is to use the definition of convexity, which states that for any two points in the set, all points along the line segment connecting them must also be in the set.

4. Are there any other properties of convex sets?

Yes, there are several other properties of convex sets, including the fact that the intersection of convex sets is also convex, and that the convex hull of a set is the smallest convex set containing all points in the original set.

5. Why is convexity important in mathematics and science?

Convexity plays an important role in optimization problems, as convex sets have nice properties that make it easier to find optimal solutions. It is also used in many other areas of mathematics and science, including geometry, economics, and computer science.