Barycenter Euclidean Space I have never seen Barycenter coordinates before, so this is a new topic to me. In some reading I found the statement : 
"In the Euclidean case the Barycenter of points $x_1,\ldots x_p$ with Barycentric coordinates $\lambda_1 \ldots \lambda_p$ is the minimizer of 
$$x\to \sum_{i=1}^p \lambda_i |x-x_i|^2. "$$
Can anyone explain what this means I'm really lost? My thoughts : $x_1,\ldots x_p$ are points in the Euclidean space (say $\mathbb{R}^d),$ under some change of coordinate system these become $\lambda_1,\ldots \lambda_p$, then this minimiser is the point which is the least distance (measured against all the points) w.r.t the distence in the new coordinate system? Like a center of mass? Help !:) 
 A: I don't think there's any change of coordinate system involved here.   
Read this e.g.
Barycentric coordinates on triangles
You can also read the more general definition above.  
Here we're talking of 3 points given (the vertices of a triangle) and how any other point can be expressed in terms of these three points.    
These $\lambda_i$ (the Barycentric coordinates) are just 3 numbers really which uniquely identify a point if some other points (the vertices of a triangle in this case) are given.   
Considering the statement you mention, you may want to post a broader part of the text. It should be something rather obvious.    
Check this too, it's an even more elementary explanation.
https://www.cut-the-knot.org/triangle/barycenter.shtml 
Basically if you assume that in each vertex of the triangle ABC you have mass of 1, then the barycentric coordinates e.g. of the centroid are (1/3, 1/3, 1/3),
and the barycentric coordinates of a mid-point (e.g. of AB) are (0, 1/2, 1/2). But if you have different masses in each vertex then you need to weight these accordingly.    
This is all related to vectors, if you know vectors you should grasp this too.  
