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.
, having a domain A and an ordered codomain B that are both closed under addition, with the following property:
An example is the square root function, having the non-negative real numbers as domain and codomain: since
Note that while a concave sequence is subadditive, the converse is false.
A useful result pertaining to subadditive sequences is the following lemma due to Michael Fekete.
By infinitary pigeonhole principle, there exists a sub-subsequence
, whose indices all belong to the same residue class modulo
This sequence, continued for long enough, would be forced by subadditivity to dip below the
The analogue of Fekete's lemma holds for superadditive sequences as well, that is:
Though we don't have continuous variables, we can still cover enough integers to complete the proof.
[2] 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.
[3][4] Besides, analogues of Fekete's lemma have been proved for subadditive real maps (with additional assumptions) from finite subsets of an amenable group [5] [6] ,[7] and further, of a cancellative left-amenable semigroup.
Entropy plays a fundamental role in information theory and statistical physics, as well as in quantum mechanics in a generalized formulation due to von Neumann.
Additionally, entropy in physics satisfies several more strict inequalities such as the Strong Subadditivity of Entropy in classical statistical mechanics and its quantum analog.
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.
Economies of scale are represented by subadditive average cost functions.
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.
The lack of subadditivity is one of the main critiques of VaR models which do not rely on the assumption of normality of risk factors.
is, assuming that the mean portfolio value variation is zero and the VaR is defined as a negative loss,
is the inverse of the normal cumulative distribution function at probability level
is the linear correlation measure between the two individual positions returns.
and, in particular, it equals the sum of the individual risk exposures when
which is the case of no diversification effects on portfolio risk.
Subadditivity occurs in the thermodynamic properties of non-ideal solutions and mixtures like the excess molar volume and heat of mixing or excess enthalpy.
In combinatorics on words, a common problem is to determine the number
The expected length of the longest common subsequence is a super-additive function of