In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots. Additive functions are special cases of subadditive functions.
for all m and n.
- Fekete's Subadditive Lemma: For every subadditive sequence , the limit exists and is equal to . (The limit may be .)
The analogue of Fekete's lemma holds for superadditive functions as well, that is: (The limit then may be positive infinity: consider the sequence .)
There are extensions of Fekete's lemma that do not require the inequality (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.
If f is a subadditive function, and if 0 is in its domain, then f(0) ≥ 0. To see this, take the inequality at the top. . Hence
The negative of a subadditive function is superadditive.
Subadditivity is an essential property of some particular cost functions. It is, generally, a necessary and sufficient condition for the verification of a natural monopoly. It implies that production from only one firm is socially less expensive (in terms of average costs) than production of a fraction of the original quantity by an equal number of firms.
In general, the price of goods (as a function of quantity) must be subadditive. Otherwise, if the sum of the cost of two items is cheaper than the cost of the bundle of two of them together, then nobody would ever buy the bundle, effectively causing the price of the bundle to "become" the sum of the prices of the two separate items.
- Fekete, M. "Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit. ganzzahligen Koeffizienten." Mathematische Zeitschrift 17 (1923), pp. 228–249.
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- Michael J. Steele (2011). CBMS Lectures on Probability Theory and Combinatorial Optimization. University of Cambridge.
- Schechter, Eric (1997). Handbook of Analysis and its Foundations. San Diego: Academic Press. ISBN 126227608., p.314,12.25
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