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The proof of this theorem is by contradiction. Suppose for convex sets S and T there are elements a and b such that a and b both belong to S∩T, i.e., a belongs to S and T and b belongs to S and T and there is a point c on the straight line between a and b that does not belong to S∩T. This would mean that c does not belong to one of the sets S or T or both. For whichever set c does not belong to this is a contradiction of that set's convexity, contrary to assumption. Thus no such c and a and b can exist and hence S∩T is convex.

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โˆ™ 2010-09-11 00:06:16
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Q: Intersection of two convex set is convex?
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Related questions

Two lines with intersection is the empty set?

It can be if the set consists of convex shapes, for example.


Is the union of two convex sets a non-convex set?

the union of two convex sets need not be a convex set.


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You normally do not have an intersection of only one set. The intersection of a set with itself is the set itself - a statement that adds little value. The intersection of two sets is the set which contains elements that are in each of the two sets.


Prove that intersection of two convex polgons is a convex as well?

I will prove a more general theorem from which your answer follows immediately. Theorem: The intersection of any number (including 2) of convex polygons is convex.ProofLet C be the intersection of Ci which is a set of iconvex polygons. By definition of intersection, if two points A and B belong to C then they belong to every one of the Ci . But the convexity of each of the Ci tells us that line segment AB is contained in Ci . Therefore, the line segment AB is in C and because ABwas arbitrary we conclude that C is convex


What is the union intersection set?

Given two or more sets there is a set which is their union and a set which is there intersection. But, there is no such thing as a "union intersection set", as required for the answer to the question.


What is the maximum possible finte number of intersection points of two non-convex quadrilaterals?

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What does intersection mean mathiticaly?

The intersection is the set of solutions that satisfy two or more mathematical expressions.


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the other one is intersection


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The answer depends on how it is halved. If the plane is divided in two by a step graph (a zig-zag line) then it will not be a convex set.


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The intersection of two distinct planes is a line. The set of common points in the line lies in both planes.


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The intersection of two sets S and T is the set of all elements that belong to both S and T.


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No. It can be infinite, finite or null. The set of odd integers is infinite, the set of even integers is infinite. Their intersection is void, or the null set.


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The concept of closure: If A and B are sets the intersection of sets is a set. Then if the intersection of two sets is a set and that set could be empty but still a set. The same for union, a set A union set Null is a set by closure,and is the set A.


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What is intersection in algebra?

A point or set of numbers that is a solution to two equations.


What is an example of an intersection in math?

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Is a circle a convex set?

no