Convex Combination Meaning I studied the formal definition of a convex combination ( the one considering more than 2 points) , but i have not understood the intuitive meaning of it.
Could you explain me ?
Thanks
 A: Well, let's look at first at the definition:

Given a vectro space $X$ and points $x_1,x_2,\dots,x_n\in X$, we call convex combination every other point $x\in X$ such that:
$$x=\sum_{i=1}^n\lambda_ix_i$$
where $\lambda_1,\lambda_2,\dots,\lambda_n\geq0$ with $\sum\limits_{i=1}^n\lambda_i=1$.

So, intuitively, it is like every convex combination of the points $x_1,x_2,\dots,x_n$ is created like: we tak "some" of $x_1$ and "some" of $x_2$ etc, up to $x_{n-1}$ and the rest (if) needed, is taken from $x_n$. Now, let's have a look at an example - we will stay at a very intuitive level, formal proofs are not difficult, but they require a lot of calculations.
Let $X=\mathbb{R}^2$ and let $A,B,C,D,E$ be the following points on the plane:

Now, a convex combination of these five points can be any point that belongs in the minimum convex set that contains all of these points - this is called "convex hull" of the points $A,B,C,D,E$. But which is this set?
Well, this is a convex set that contains all of the points, but one could do better than this with not that much effort:

If we elaborate a little bit on this, we can see that, a way to find this set is as follows:

*

*Find the point that is "lower" than any other (the smallest $y$-coordinate). In our case this is $C$.

*Pin a rope onto this point and start moving this rope around the plane counterclockwise starting from a horizontal position, until you meet $C$ again. If one can think of the other points as "pegs", then the following shape occurs:

Now, every point into this shape - or its boundary - is a convex combination of $A,B,C,D,E$. Actually, since $E$ is in the inner part of the convex hull, it is a convex combination of $A,B,C,D$, so, these four points are sufficient to generate this shape - our rope did not meet the $E$ "peg" while rotating. A step further, one can show that for every $x$ in the convex hull described, we can find $3$ of $A,B,C,D$ such that $x$ is a convex comabination of them (these three points will form a triangle that includes the point $x$, from wichi it is also implied that the tringle is the elementary shape of the Eucledian plane):

Note: If we let $\lambda_i\in\mathbb{R}$, $i=1,2,\dots,n$ with $\sum\limits_{i=1}^n\lambda_i=1$, we can get every point on the plane that these points create - in our case this is the same as $\mathbb{R}^2$. This kind of combination is called an "affine" combination of $A,B,C,D,E$.
