Math Problem Statement
How do you prove that the set S= {(x,y) |x²+y²|<=4} is a convex set?
Solution
To prove that the set is convex, we need to show that for any two points and in the set , every point on the line segment joining and is also in the set .
Step-by-Step Proof:
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Restating the Problem:
The set is the interior (including the boundary) of a circle with radius 2 centered at the origin. This means that for any point , it holds that . -
Convexity Definition:
A set is convex if, for any two points and , the line segment joining and , defined by the parametric equation
for , is also contained in . -
Parametric Point on the Line Segment:
Consider the point on the line segment . -
Check if Point Lies in :
To check if the point lies in , we need to verify whether it satisfies the condition for being in , which is
Let's compute :
By expanding the terms, we get:
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Use the Condition for and :
Since and , we know that and . Therefore, we can bound the above expression as follows: This simplifies to: -
Final Verification:
The term , so we get: The second term does not exceed 4 because and , implying that .
Conclusion:
Since for all , every point on the line segment between and remains inside the set . Thus, is convex.
Do you have any questions or would you like further details on any part of the proof?
Here are 5 related questions:
- What is the geometric interpretation of convexity in higher dimensions?
- How can you prove convexity for more complex sets, like ellipses or polygons?
- Can a set that contains holes or gaps still be convex?
- How does convexity relate to optimization problems in mathematics?
- How would you prove that the set (the boundary of the circle) is not convex?
Tip: When proving convexity, checking line segments between points within the set is a fundamental method.
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Math Problem Analysis
Mathematical Concepts
Convexity
Geometry
Algebra
Formulas
x² + y² ≤ 4
Parametric equation for a line segment: P(t) = (1 - t)A + tB
Theorems
Definition of Convex Sets
Suitable Grade Level
College Level (Advanced Geometry and Linear Algebra)