Prove S is Convex: Help with Convexity

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In summary, the conversation discusses proving the convexity of a set S in R^n, defined as {(X1, ..., Xn) € R^n | Xi ≥ 0, X1 + ... + Xn = 1}. The conversation also introduces two functions, f and g, and discusses their relationship to S. The speaker is unsure of how to proceed with the proof, but the listener suggests going back to the definition of a convex set and showing that the combination of two points in S is still in S.
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Shaad
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How do i prove this?

Let S = {(X1, …., Xn) € R^n | Xi ≥0, X1 + … +Xn = 1}. Show that S is convex.

Suppose f(Si) = {X1, X2,..., Xn}
AND g(Si) = {X1 + X2+ ...+Xn}

but If do that, then f(Si) and g(Si), both will increase. Now I'm not sure where to go from here.

*PS: Both functions are continuous convex function of Xi (Where = 1, 2, ..., N)

Now as i mentioned earlier, both function increases, so does that also mean that S is convex?
 
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  • #2
Your notation is confusing. What is Si, and what is the target of the function f? What are you trying to do?

Anyway, the way I'd do it is just to go back to the definition of a convex set. If X=(X1,...,Xn) and Y=(Y1,...,Yn) are in S, then show sX+(1-s)Y is in S for any s in [0,1]. This direct approach is easy enough in this situation.
 

FAQ: Prove S is Convex: Help with Convexity

What is the definition of convexity?

Convexity is a property of a set or function that describes the shape of its graph. A set or function is convex if every line segment connecting two points on the graph lies entirely within the set or above the function.

How do you prove that a set is convex?

To prove that a set is convex, we can use the definition of convexity and show that every line segment connecting two points in the set lies entirely within the set. This can be done by choosing two arbitrary points in the set and showing that the line connecting them is also within the set.

What are some properties of convex sets?

Convex sets have the property that any point on a line segment connecting two points in the set is also within the set. They also have the property that the intersection of any number of convex sets is also convex.

How can convexity be applied in real-world problems?

Convexity has many applications in fields such as economics, optimization, and machine learning. In economics, convexity is used to describe the preferences of consumers and to analyze market equilibrium. In optimization, convexity is used to find the global minimum of a function. In machine learning, convexity is used to design and analyze convex loss functions for training models.

What are some common examples of convex sets?

Some common examples of convex sets include circles, ellipses, polygons, and closed intervals on a number line. In higher dimensions, convex sets can also include spheres, cubes, and convex polyhedra.

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