Maximum-minimums identity

In mathematics, the maximum-minimums identity is a relation between the maximum element of a set S of n numbers and the minima of the 2n − 1 nonempty subsets of S.

Let S = {x1, x2, ..., xn}. The identity states that

\begin{align} \max\{x_1,x_2,\ldots,x_{n}\} & = \sum_{i=1}^n x_i - \sum_{i<j}\min\{x_i,x_j\} +\sum_{i<j<k}\min\{x_i,x_j,x_k\} - \cdots \\ & \qquad \cdots + \left(-1\right)^{n+1}\min\{x_1,x_2,\ldots,x_n\},\end{align}

or conversely

\begin{align} \min\{x_1,x_2,\ldots,x_{n}\} & = \sum_{i=1}^n x_i - \sum_{i<j}\max\{x_i,x_j\} +\sum_{i<j<k}\max\{x_i,x_j,x_k\} - \cdots \\ & \qquad \cdots + \left(-1\right)^{n+1}\max\{x_1,x_2,\ldots,x_n\}. \end{align}

For a probabilistic proof, see the reference.

See also

References

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