In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two
elements
Element or elements may refer to:
Science
* Chemical element, a pure substance of one type of atom
* Heating element, a device that generates heat by electrical resistance
* Orbital elements, parameters required to identify a specific orbit of ...
of the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
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.
Definitions
A subadditive function is a
function , having a
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
''A'' and an
ordered codomain ''B'' that are both
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
under addition, with the following property:
An example is the
square root function, having the
non-negative real numbers as domain and codomain,
since
we have:
A
sequence , is called subadditive if it satisfies the
inequality
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.
Note that while a concave sequence is subadditive, the converse is false. For example, randomly assign
with values in
, then the sequence is subadditive but not concave.
Properties
Sequences
A useful result pertaining to subadditive sequences is the following
lemma
Lemma may refer to:
Language and linguistics
* Lemma (morphology), the canonical, dictionary or citation form of a word
* Lemma (psycholinguistics), a mental abstraction of a word about to be uttered
Science and mathematics
* Lemma (botany), a ...
due to
Michael Fekete.
The analogue of Fekete's lemma holds for superadditive sequences 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
to hold for all ''m'' and ''n'', but only for ''m'' and ''n'' such that
Moreover, the condition
may be weakened 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.
Besides, analogues of Fekete's lemma have been proved for subadditive real maps (with additional assumptions) from finite subsets of an amenable group
,
and further, of a cancellative left-amenable semigroup.
Functions
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
A
concave function
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.
Definition
A real-valued function f on an in ...