Subadditive function

From Exampleproblems

In mathematics, a subadditive function is a function f(x) such that

for all x and y in the domain of f. In other words, if the area under the curve is greater when x and y are two separate curves added together than when x and y are combined and used to define a single curve.

A sequence { a_n }, n ≥ 1, is called subadditive if it satisfies the inequality

for all m and n. The major reason for use of subadditive sequences is the following lemma due to M. Fekete.

Lemma: For every subadditive sequence { a_n }, n ≥ 1, the limit lim a_n/n exists and is equal to inf a_n/n.

The analogue of Fekete's lemma holds for subadditive functions as well.

There are extensions of Fekete's lemma that do not require equation (1) to hold for all m and n. There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both superadditivity and subadditivity is present.¹

See also

• Triangle inequality

References

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Note

• Template:Ent A good exposition of this topic may be found in Steele's Probability theory and combinatorial optimization given in the references.

This article incorporates material from subadditivity on PlanetMath, which is licensed under the GFDL.