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In [[mathematics]], '''subadditivity''' is a property of a function that states, roughly, that evaluating the function for the sum of two [[element (set)|elements]] of the [[Domain of a function|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 [[norm (mathematics)|norms]] and [[square roots]]. [[Additive function]]s are special cases of subadditive functions.
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==Definitions==
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A subadditive function is a [[function (mathematics)|function]] <math>f \colon A \to B</math>, having a [[Domain of a function|domain]] ''A'' and an [[partial order|ordered]] [[codomain]] ''B'' that are both [[closure (mathematics)|closed]] under addition, with the following property:
::<math>\forall x, y \in A, f(x+y)\leq f(x)+f(y).</math>
 
An example is the [[square root]] function, having the [[non-negative]] [[real number]]s as domain and codomain,
since <math>\forall x, y \geq 0</math> we have:
::<math>\sqrt{x+y}\leq \sqrt{x}+\sqrt{y}.</math>
 
A [[sequence]] <math>\left \{ a_n \right \}, n \geq 1</math>, is called '''subadditive''' if it satisfies the [[inequality (mathematics)|inequality]]
::<math>(1) \qquad a_{n+m}\leq a_n+a_m</math>
for all ''m'' and ''n''.
 
==Properties==
A useful result pertaining to subadditive sequences is the following [[lemma (mathematics)|lemma]] due to [[Michael Fekete]].<ref>Fekete, M. "Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit. ganzzahligen Koeffizienten." ''Mathematische Zeitschrift'' '''17''' (1923), pp. 228–249.</ref>
 
:'''Fekete's Subadditive Lemma:'''  For every subadditive sequence <math>{\left \{ a_n \right \}}_{n=1}^\infty</math>, the [[Limit of a sequence|limit]] <math>\lim_{n \to \infty} \frac{a_n}{n}</math> exists and is equal to <math>\inf \frac{a_n}{n}</math>. (The limit may be <math>-\infty</math>.)
 
The analogue of Fekete's lemma holds for superadditive functions as well, that is:
<math>a_{n+m}\geq a_n + a_m.</math>  (The limit then may be positive infinity: consider the sequence <math>a_n = \log n!</math>.)
 
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 [[superadditive|superadditivity]] and subadditivity is present.<ref>Michael J. Steele. "Probability theory and combinatorial optimization". SIAM, Philadelphia (1997). ISBN 0-89871-380-3.</ref><ref>{{cite video|author=Michael J. Steele|title=CBMS Lectures on Probability Theory and Combinatorial Optimization|publisher=University of Cambridge|year=2011|url=http://sms.cam.ac.uk/collection/1189351}}</ref>
 
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. <math>f(x) \ge f(x+y) - f(y)</math>. Hence <math>f(0) \ge f(0+y) - f(y) = 0</math>
 
A [[concave function]] <math>f: [0,\infty) \to [0,\infty)</math> with ''f''(0) = 0 is also subadditive.
To see this, one first observes that <math>f(x) \ge \textstyle{\frac{y}{x+y}} f(0) + \textstyle{\frac{x}{x+y}} f(x+y)</math>.
Then looking at the sum of this bound for <math>f(x)</math> and <math>f(y)</math>, will finally verify that ''f'' is subadditive.<ref>{{cite book | last = Schechter | first=Eric | authorlink=Eric Schechter| title=Handbook of Analysis and its Foundations | publisher=Academic Press | location = San Diego | year=1997 | id=ISBN 126227608 }}, p.314,12.25</ref>
 
The negative of a subadditive function is [[superadditivity|superadditive]].
 
==Economics==
 
Subadditivity is an essential property of some particular [[Cost curve|cost function]]s. 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.
 
[[Economies of scale]] are represented by subadditive [[average cost]] functions.
 
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.
 
==See also==
*[[Triangle inequality]]
*[[Superadditivity]]
*[[Choquet integral]]
 
==Notes==
<references />
 
==References==
*[[György Pólya]] and [[Gábor Szegő]]. "Problems and theorems in analysis, volume 1". Springer-Verlag, New York (1976). ISBN 0-387-05672-6.
 
==External links==
{{PlanetMath attribution|id=4615|title=subadditivity}}
 
[[Category:Mathematical analysis]]
[[Category:Sequences and series]]

Latest revision as of 03:00, 18 December 2014

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