# Barycenter

This is the center of mass and is the point at which a mass (body) can be considered as being concentrated without altering the effect of the attraction that the earth has upon it. This is the average position of particles located at postions $LaTeX: x^i$ weighted by their masses $LaTeX: m_i$,

$LaTeX: \frac{\sum_i m_i x^i}{\sum_i m_i}.$

In mathematical programming, this is usually used to mean the simple average of the extreme points of a polytope,

$LaTeX: \textstyle \frac{1}{m} \sum_i x^i,$

where $LaTeX: \{x^1, \ldots, x^m\}$ is the set of extreme points. This could be a weighted average,

$LaTeX: \textstyle \sum_i w_i x^i,$

where $LaTeX: w \in S_m$.