Geometric mean

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Given a nonnegative LaTeX:  x \in \mathbb{R}^n and a weight vector, LaTeX: a in the simplex, LaTeX: S_n, the geometric mean is the value:

LaTeX: 
x(1)^a(1) * x(2)^a(2) * ... * x(n)^a(n).

For example, for LaTeX: a=(1/2, 1/2), the geometric mean of LaTeX: x = (1, 9) is LaTeX: 1*3=3. A fundamental inequality that provides a foundation for geometric programming is that the geometric mean is never greater than the arithmetic mean:

LaTeX: 
\Pi_j x(j)^a(j) \le \textstyle\sum_j a(j)x(j).

for all nonnegative LaTeX: x \in \mathbb{R}^n and LaTeX: a \in S_n. Further, equality holds only if LaTeX: x(j) is constant for all LaTeX: j.

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