00:02
Hello, let's have a look at the question.
00:04
So here we have to prove that any convex set s contains all the convex combinations of three points in the set.
00:12
Now, here to prove this we will use the definition of convex combination.
00:17
So here let's prove this.
00:21
Let's assume that s is a convex set and let us take that capital a and b and c be the three arbitrary points in capital s.
00:53
Now, we want to show that convex combination of a, b and c is also in s.
01:01
So here a convex combination of a, b and c is defined a point capital p that can be expressed as capital p is equals to lambda multiplied with a plus mu multiplied with b and plus gamma multiplied with c where lambda, mu and gamma are the non -negative scalars that satisfy the condition lambda plus mu plus gamma is equals to 1.
02:21
Now, here we will consider the point capital p is equals to lambda a plus mu b and gamma c.
02:42
Now, we need to show that capital p is in capital s.
02:51
Now, since we know that s is a convex set therefore, this means that any two points let it be x and y in s.
03:11
So then the line segment connecting capital x and y is also entirely contained in s.
03:28
So here we will consider that x and y are in s.
03:34
So we can choose that x is equals to a and y is equals to b...