Why is the convex combination of $3$ points in $\Bbb R^2$ a triangle and not a V-like shape?

Question

I am new to convexity. I do find it really confusing why the convex combination of $3$ vectors (points) is a triangle and not a V-like shape. With $2$ vectors, it is a segment then why is the case of $3$ vectors it is a triangle or for vectors greater than $2$ there is a shape i am talking about general case not extreme case where all $3$ vectors are on a line or something like that.

Answer 1

Let $a,b,c$ be the three points in our $\mathbb{R}^2$. As you pointed out we can form segments between 2 points by taking the convex combinations of the two points, e.g. $\lambda a+(1-\lambda)b$. And then considering the $\mu b + (1-\mu)c$ gives you your "V". But of course in the convex combination of the three points, e.g. $\alpha_1 a +\alpha_2 b + \alpha_3 c$ you also have the segment $\nu c + (1-\nu) a$ which gives you the "last line of your "V" to complete your triangle".

Answer 2

Let your three points be represented by the vectors $a,b,c$.

The convex combination of the points $a$ and $b$ are all the points of the form $xa+yb$ where $x$ and $y$ are non-negative reals which sum to $1$. In other words they are all the points on the line segment between $a$ and $b$.

Similarly you obtain all the points on the other edges of the triangle.

Any point in the interior of the triangle is on a line between two points on edges and so they also are in the convex c0mbination of the three points.