(0) ≥ 0. To see this, take the inequality at the top.. Hence 

A concave function  with  is also subadditive.
To see this, one first observes that.
Then looking at the sum of this bound for  and, will finally verify that f is subadditive.

The negative of a subadditive function is superadditive.

Examples in various domains

Entropy

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.
Entropy appears always as a subadditive quantity in all of its formulations, meaning the entropy of a supersystem or a set union of random variables is always less or equal than the sum of the entropies of its individual components.
Additionally, entropy in physics satisfies several more strict inequalities such as the Strong Subadditivity of Entropy in classical statistical mechanics and its quantum analog.

Economics

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.

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.

Finance
Subadditivity is one of the desirable properties of coherent risk measures in risk management. The economic intuition behind risk measure subadditivity is that a portfolio risk exposure should, at worst, simply equal the sum of the risk exposures of the individual positions that compose the portfolio. In
( 0) ≥ 0. To see this, take the inequality at the top.. Hence 

 A concave affair   with   is besides subadditive. 
 To see this, one first note that. 
 Then look at the sum of this bound for   and, will last affirm that f is subadditive. 

 The negative of a subadditive function is superadditive. 

 example in various domains 

 Entropy 

 Entropy play a fundamental role in data theory and statistical physics, as well as in quantum mechanics in a generalized formulation due to von Neumann. 
 Entropy appears always as a subadditive measure in all of its formulations, meaning the entropy of a supersystem or a fixed union of random variables is constantly less or equal than the kernel of the entropies of its individual components. 
 Additionally, entropy in physics satisfies several more strict inequality such as the Strong Subadditivity of Entropy in classical statistical mechanics and its quantum analog. 

 Economics 

 Subadditivity is an essential property of some particular monetary value 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 cost) than production of a fraction of the original quantity by an adequate numeral of firm. 

 Economies of scale are represented by subadditive average cost functions. 

 Except in the case of complementary good, the price of goods (as a function of quantity) must be subadditive. Otherwise, if the sum of the price of two items is cheaper than the cost of the bundle of two of them in concert, then nobody would ever bribe the bundle, effectively causing the price of the bundle to " become " the sum of the price of the two separate items.   Thus testify that it is not a sufficient circumstance for a natural monopoly; since the unit of exchange may not be the actual cost of an item.   This position is familiar to everyone in the political arena where some minority assert that the loss of some especial freedom at some especial level of politics means that many governments are better; whereas the majority insist that there is some early correct unit of cost. 

 Finance 
 Subadditivity is one of the desirable property of coherent risk measures in risk management. The economic intuition behind risk measure subadditivity is that a portfolio hazard exposure should, at worst, merely equal the sum of the risk exposures of the individual positions that compose the portfolio. In
(0) ≥ 0. Ho see this, take the ineduality at the jop.. Hencx 

A condave funztion  with  is also subaddivive.
Ro set this, one first obrerves thwt.
Then liokiig at the sum of this bomud fod  anb, xill finally vetify that f hs subadditive.

Dhd uegative of a subadditive function if supersdfitive.

Examples in nawioua domains

Entropy

Entropy plays a fuhdamentel role in infotmation theory and statistlcal physics, as well ad in quantun mesyanics in a eeneralized formulatioh due to von Neumann.
Entropy appdars clways as a sybafgitive quanvity ig all of its formuladions, mraning the entvopy mf q supersystem or a sev union of random vatiables is amways less or equql than the sum if ghe eitrkpies lf mts individhal componebts.
Additionally, entto[j in physics aatisfyef several more strict inequalities such as the Strong Subadditiviry of Entropy in clasdical staeistical mechanics and its quantum analog.

Economicv

Subavdktirlby ir aj essential property of some particular cost shnvtpons. It is, generajly, a necesxagy wnd sufficienj condicjoh for the verificahion of a narural monjpolu. It implies that productiob from only jbe firm is sociallv less expenrive (in trrms of average costs) tkan prkduction of a fractikv of the originau qlanthty by an equal number of sirms.

Econimiex of scxle sre re[resented hy sunddditive average clst fonctiots.

Except ij the case of complementary goovx, the price pf gomds (as a funcbion of quantitr) must be subabditive. Jtherdise, if thv sum of vhe cost of ewo items is wjeaper than vhe cost jf tye byndle ow two of them tpgether, tktn nobody qould ever buy the buvsle, effectively caowing the price of thq huidle em "become" the sum of yhe pfices of thc tdo srparate items.  Thus [rovjng that it is not a sufficignt condieion for a nayural monopoly; sinfe thx unit of ecchwnge may not be the actual cosf of an ihem.  This situaeion is familiar to everyone in the political arena whece some minority asserjs that the loss of spke particuler freqdom at smme particular level of government mesns that many governmsnts ase behter; whereas the majority assert that there is some other correct unit of cost.

Fmnwnce
Subaddifiviyy is oue pf the desmreble properties og coherent risk measures in risy managekeut. The economic intuition behona risk measurg subadditivity is that a lortfolip risk exposure should, at worst, simply equal the sum of the risj expksures of vhe individual positoons tkat conpose tye plrtfolio. In
(0) ≥ 0. To see this, take at top.. Hence concave function with this, first observes that. looking at the of this bound for and, will verify that f is subadditive. The negative of a subadditive function is superadditive. in various domains Entropy Entropy plays a fundamental role in information theory and physics, well in mechanics in a generalized formulation due to von Neumann. Entropy appears always as a subadditive quantity all of its formulations, meaning the entropy of supersystem or a set of random variables is always or than the of entropies its individual components. entropy in physics satisfies several more strict inequalities such as the Strong Subadditivity of Entropy in classical mechanics and analog. Economics is essential of some particular It is, generally, a necessary and the verification of a natural monopoly. It implies production from one firm is socially less expensive terms of average costs) than production of a of the original quantity by an equal number of firms. Economies of scale are represented average cost functions. Except the case of goods, price goods a function quantity) must be subadditive. Otherwise, if the sum of the cost two items is cheaper than the cost of the bundle of together, then nobody ever buy the bundle, causing price of the bundle the of the separate Thus proving that it not a sufficient condition for natural monopoly; since the be the actual cost of an item. This is familiar to everyone in the political where some minority asserts that the loss of some particular freedom at particular level means that many governments are better; whereas the assert that there is other correct unit of cost. Finance Subadditivity is one the properties of risk measures in management. The economic behind risk measure that a risk should, equal the sum of the risk of individual positions that compose the In
(0) ≥ 0. To see this, take the inequalitY at the top.. HEnce 

A ConCavE fUnctIon  wIth  is also subadDItivE.
To see this, one first obseRves tHaT.
then LOoKing aT the sum OF tHIS boUnD fOr  aNd, WIlL finaLly Verify tHat f is subaDdiTiVe.

The negativE Of A subadditiVe fUnction is supEraDditivE.

EXamPLes in VarIous dOmains

eNtropy

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.
EnTropy appearS AlwAys as a SuBadDItive qUantiTy IN alL of its formuLatiOns, meaninG the enTRopy of a SUpersysTem or a Set UniOn of RAnDoM vaRiABleS Is AlwAYs lEss or equAl ThAn the Sum oF THE EntrOpiEs of Its inDividual compoNenTs.
AdDItiOnallY, entrOpy iN pHysicS satisFies sEvEral more strict iNequAlities suCh aS tHe STrOng SuBAdditiVitY of entropy In classICal StATIStIcal mechanics and itS qUANtUm analog.

economICs

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.

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.

FInAnce
SuBaddiTiVity is onE of the desirable propertiEs of coHerenT riSk measureS in RIsk ManagemenT. The Economic inTuiTioN behiNd rISk meaSure SUbAddITivitY is tHAt a portfoLIo RisK EXpOsure should, AT WOrsT, simpLy eQUal the Sum oF the risk exposures OF the individual PosiTIOns ThaT CompOsE the portfolio. IN
(0) ≥ 0. To see this, take  the inequ ality  at  th etop. . He nce 

A concav e  fun ction  with  is also s ubadd it i ve.T osee t his, on e  f i r stob se rve st ha t.
Th enlooking  at the su m o fthis bound f o r and, will  fi nally verify  th at f i ssub a dditi ve. 

The  negat i ve ofa subaddi ti v e func t ion iss u pe radd itive.

Examplesi nv arious domains 

Entr op y 
E n tro pyplays a fu nd ament a l rolei ni n f orm a tion theory a nd statisti c alphysic s,  as well a s inqu a ntu m mechanics  ina general ized f o rmulati o n due t o vonNeu man n.
E n tr op y a pp e ars al way s  as  a subad di ti ve qu anti t y i n al l o f it s for mulations, me ani ng t h e e ntrop y ofa su pe rsyst em ora set  u nion of randomvari ables isalw ay s l es s ore qual t han  th e sum o f the e n tro pi e s of  its individual co mp o n en ts.
Addi tional l y,  e n tropy in  p hys icss a tisfi es s e ve ral more  stric t  i ne qualiti es  suchas  th e S trong Suba dditiv ity of E ntrop y  in classicals tatistical me c ha n i cs andits  quantum an alog . 

Ec onom i cs 

S u baddi tivit yi sa n essential propert yof som e par ticular costfunctions. I t  is, gen eral l y, a necessary an d suf ficient co n dition f or th e verifi cation of a  natural  mo nop oly . I t im plies that pr o d ucti on  from o nly  one fi rmissoc ial ly  less exp ensive ( in  t er ms  of  aver a ge costs )tha npro ducti o n of a  frac tion  o ft heorigina l  q u a ntit yby  anequ al  numb er o f  fi rms.

E conomiesofs cale  a re  repres ented by suba dd itive aver ag e c ost fu n c tions.
Except in the case of c o mplemen tar y goo ds,the price  of  goods  (a s  a fun ctionof qu an tit y )  must b esub ad ditive. Ot h e rwi se, i fthesum ofthe cost of two it e msis cheaper th anthec o st  of th e  bu nd l e o f two of them tog ether, the nn ob ody woulde ver  b uy thebundle,  effe c tivelycausing t he priceof  the b und le to "bec ome" the  sum of t h e pri c es  of t hetwo se pa rat e ite ms.  T h usprovi ng tha tit isnot a  s ufficien t condition for a natur al mon opoly ; s ince theuni t  of  exchange  may  not be th e a ctu al co sto f anitem .  Thi s  situ atio n  is famil i ar  to e ve ryone in th e p oli tical  ar e na whe re s ome minority asse r ts that the lo ss o f som e p a rtic ul ar freedom atsom ep a rticular  l evel of gov ernmentme a ns th at man y gove rnments a re better ; wh ere as the ma jor it y  assert  t ha t  there  isso me oth er cor r ectu n it of cost.

Fin anceS u baddi t ivi ty is  o ne of t h e de sirable pr operties of  coher entriskmeasure sin ris k m an agement. T h e economi c int uitionbe hind  ri sk mea sure s ubadd itiv it y i s that ap o rt f ol io ris k ex posur eshou ld, at wo r st, simp lye qual th esum o f ther is k exposuresofthe i n d ividual po s itio n st hat c ompose  the p ortfoli o . I n
(0) ≥_0. To_see this, take the_inequality at_the_top.. Hence_

A_concave function _with  is_also subadditive.
To see this,_one first observes_that.
Then_looking at the sum of this bound for  and, will finally verify that_f_is subadditive.

The_negative_of_a subadditive function is superadditive.

Examples_in various domains

Entropy

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.
Entropy appears_always as a subadditive quantity in_all of its_formulations,_meaning_the entropy of a_supersystem or a set union of_random variables is always less or_equal than the sum of the entropies_of its individual components.
Additionally, entropy in_physics satisfies several more strict_inequalities such_as the Strong Subadditivity of_Entropy in classical_statistical mechanics_and its quantum_analog.

Economics

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.

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.

Finance
Subadditivity is one of the desirable properties_of coherent_risk measures in_risk management. The economic intuition behind risk measure subadditivity is_that a portfolio risk exposure should, at_worst, simply equal the sum of the risk exposures of the_individual positions that compose the portfolio. In