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