Suvarna Garge (Editor)

Maximum minimums identity

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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

max { x 1 , x 2 , … , x n } = ∑ i = 1 n x i − ∑ i < j min { x i , x j } + ∑ i < j < k min { x i , x j , x k } − ⋯ ⋯ + ( − 1 ) n + 1 min { x 1 , x 2 , … , x n } ,

or conversely

min { x 1 , x 2 , … , x n } = ∑ i = 1 n x i − ∑ i < j max { x i , x j } + ∑ i < j < k max { x i , x j , x k } − ⋯ ⋯ + ( − 1 ) n + 1 max { x 1 , x 2 , … , x n } .

For a probabilistic proof, see the reference.

References

Maximum-minimums identity Wikipedia
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