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 maps are special cases of subadditive functions.
for all m and n. This is a special case of subadditive function, if a sequence is interpreted as a function on the set of natural numbers.
- 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, but only for m and n such that Moreover, the condition may be weaken as follows: provided that is an increasing function such that the integral converges (near the infinity).
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.
- Theorem: For every measurable subadditive function the limit exists and is equal to (The limit may be )
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.
Examples in various domains
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.
Except in the case of complementary goods, 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. Thus proving that it is not a sufficient condition for a natural monopoly; since the unit of exchange may not be the actual cost of an item. This situation is familiar to everyone in the political arena where some minority asserts that the loss of some particular freedom at some particular level of government means that many governments are better; whereas the majority assert that there is some other correct unit of cost.
- Fekete, M. (1923). "Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten". Mathematische Zeitschrift. 17 (1): 228–249. doi:10.1007/BF01504345.
- de Bruijn, N.G.; Erdös, P. (1952). "Some linear and some quadratic recursion formulas. II". Nederl. Akad. Wetensch. Proc. Ser. A. 55: 152–163. (The same as Indagationes Math. 14.) See also Steele 1997, Theorem 1.9.2.
- Michael J. Steele. "Probability theory and combinatorial optimization". SIAM, Philadelphia (1997). ISBN 0-89871-380-3.
- Michael J. Steele (2011). CBMS Lectures on Probability Theory and Combinatorial Optimization. University of Cambridge.
- Hille 1948, Theorem 6.6.1. (Measurability is stipulated in Sect. 6.2 "Preliminaries".)
- Schechter, Eric (1997). Handbook of Analysis and its Foundations. San Diego: Academic Press. ISBN 0-12-622760-8., p.314,12.25
- György Pólya and Gábor Szegő. "Problems and theorems in analysis, volume 1". Springer-Verlag, New York (1976). ISBN 0-387-05672-6.
- Einar Hille. "Functional analysis and semi-groups". American Mathematical Society, New York (1948).
- N.H. Bingham, A.J. Ostaszewski. "Generic subadditive functions." Proceedings of American Mathematical Society, vol. 136, no. 12 (2008), pp. 4257–4266.