EXTREMAL PROPERTIES OF HALF-SPACES FOR LOG-CONCAVE DISTRIBUTIONS. BY S. BOBKOV Syktyvkar University

Transcription

1 Te Aals of Probability 1996, Vol. 24, No. 1, 3548 EXTREMAL PROPERTIES O HAL-SPACES OR LOG-CONCAVE DISTRIBUTIONS BY S. BOBKOV Syktyvkar Uiversity Te isoperimetric problem for log-cocave product measures i equipped wit te uiform distace is cosidered. Necessary ad sufficiet coditios uder wic stadard alf-spaces are extremal are preseted. 1. Itroductio. Let be a probability measure o te real lie. Cosider te value Ž 1.1. r p. if Ž A D., were is te product measure i, D, 1 is te -dimesioal cube i ad te ifimum is te over all Borel-measurable sets A of measure Ž A. p; 0 p 1, 0. We are searcig for ecessary ad sufficet coditios uder wic te ifimum i 1.1 is attaied at te alf-spaces A x : x c4 1 of measure p, for all p ad Žte alf-spaces of tis form will be called stadard.. I tese otes, we give suc coditios i te case were is log-cocave. Tere exists te followig probabilistic cosequece of te above property. If 1,..., are idepedet radom variables o some probability space Ž,,., wit commo distributio, cosider a family of radom variables SŽ t., idexed by a set T ad formed by liear combiatios, Ý i i SŽ t. a Ž t., were te coefficiets ai are arbitrary fuctios o T suc tat sup a Ž t. 1. Ý 1 i t i Set M sup SŽ t. t. Te te extremal property of te stadard alf-spaces i te isoperimetric problem Ž 1.1. implies tat, for all p Ž 0, 1. ad 0, 4 4 Ž 1.2. M m Ž M. m Ž., p 1 p 1 were m p deotes quatile of order p of a radom variable. or 0, te coverse iequality sould be writte i Ž I oter words, te deviatios i Received July 1993; revised Marc Tis researc was partially supported by te NS, Air orce Office of Scietific Researc Grat ad Army Researc Office Grat DAAL G AMS 1991 subject classificatio. 60G70 Key words ad prases. Isoperimetry 35

2 36 S. BOBKOV of M from its quatiles are ot larger ta tose of te origial radom variable 1; tis is also equivalet to existece of a Lipscitz fuctio from to suc tat M ad Ž. 1 are idetically distributed. I terms of te isoperimetric problem, property Ž 1.2. ca be equivaletly expressed as follows: Ž 1.1. attais a miimum at te stadard alf-spaces witi te class of all covex subsets of. We will prove, owever, tat te last implies te extremal property of tese alf-spaces witi te class of all Borel-measurable subsets of. Moreover, i order to obtai suc a extremal property wit respect to te supremum distace, we will prove tat it oly suffices to assume tat Ž 1.2. is fulfilled for two radom variables Ži te case 2.: M ad M max, I te case were is a Gaussia measure o te real lie, te alf-spaces possess a muc more itrisic property 6, 4 ; tey are extremal i Ž 1.1. eve if D is replaced by te uit Euclidea ball i. Likewise, Ž 1.2. olds uder 2 te weaker assumptio sup Ý a Ž t. 2 2 t i i 1. It ca be sow tat suc a strog property Ž we 2. caracterizes Gaussia measures i te class of all probability distributios. Talagrad 7 cosidered a elargemet of sets A wic is, i a certai sese, eve smaller ta te Euclidea -eigborood of A. A iequality e proved for te two-sided expoetial distributio does ot express ay extremal property but states a rater strog variat of te so-called cocetratio-of-measure peomeo. Iequalities Že.g., as i ad 2. for te uiform elargemet defied by te supremum distace state te weakest variat of tis peomeo. Deote by te distributio fuctio of te measure : Ž x. ŽŽ, x., x. By defiitio, is log-cocave if it as a desity f suc tat te fuctio logž f. is cocave o te iterval Ž a, b., were 4 4 a if x : Ž x. 0, b sup x : Ž x. 1. I geeral, a b. Necessarily, f is cotiuous ad positive o a, b, icreases o a, b, ad so a iverse : Ž 0, 1. Ž a, b. exists. I tis paper, we preset te followig statemet o log-cocave measures. THEOREM 1.1. Let 2. Te stadard alf-spaces are extremal for all 0 ad p Ž 0, 1. i te isoperimetric problem Ž 1.1. if ad oly if te followig old: Ž a. is symmetric aroud a poit Ž wic is te media of.; Ž b. te support of is te real lie Ž i.e., a, b. ; Ž c. for ay p, q Ž 0, 1., f Ž pq. f Ž p. f Ž q. Ž pq p q Tis statemet does ot deped o te dimesio. Note tat te equality Ž I p f Ž p.. defies 1 correspodece betwee te family of all

3 EXTREMAL PROPERTIES 37 distributio fuctios wic ave a eve positive cotiuous desity f o Ž a, b. Ž a b. ad te family of all positive cotiuous fuctios I o 1 Ž 0, 1. symmetric aroud. I additio, Ž 1.4. b dpi Ž p.. 2 H 1 12 Te log-cocavity of f is equivalet to cocavity of I Ž see Propositio 6.1., so, uder assumptios Ž a. ad Ž b., Ž 1.3. determies a certai subset of te set 1 of all cocave, positive, symmetric aroud fuctios o Ž 0, 1. 2 for wic te itegral Ž 1.4. is ifiite. Ofte, it is ot easy to ceck Ž We will give te followig weaker coditio: te fuctio logž f. is cocave. or te stadard Gaussia measure, te last property i.e., te secod derivative of logž f. is ever positive follows from te iequality 1 Ž x. xfž x. Ž1 x 2., x 0. Cosider aoter example. Let Ž x. 1Ž1 expž x.., x Žso-called logistic distributio fuctio.. Te, te fuctio logž fž x. Ž x.. logž1 x. Ž e is also cocave. I tis case, f Ž p.. pž 1 p., ad Ž 1.3. is easy to verify. Te two-sided expoetial distributio, of desity fž x. expž x. 2, x, is very close to te above metioed distributio, but Ž 1.3. is Ž ot fulfilled for oe ca take p q 0.7. I tis case, f Ž p.. mi p, 1 p 4. Tus, oe ca say tat te extremal property of alf-spaces is ot determied by te tail beavior of ; it meas a quality of some differet ad delicate ature. 2. Descriptio of te proof. or te reader s coveiece, we first ote several steps wic will be performed to prove Teorem 1.1. Te first step is to solve te oe-dimesioal isoperimetric problem. A special case were as desity fž x. expž x. 2, x, as bee studied i 7. I Sectio 3 we will prove by similar metods te followig statemets. PROPOSITION 2.1. Let be a log-cocave measure o te real lie. Te for all p Ž 0, 1., 0, te value Ž A,. is miimal o te class of all Borel-measurable sets A of measure Ž A. p, if A is te iterval Ž, a or te iterval b,. of -measure p Ž i bot cases.. PROPOSITION 2.2. or a log-cocave measure o te real lie, te itervals Ž, a are extremal for all p Ž 0, 1., 0, if ad oly if is symmetric aroud its media. Sice i Propositio 2.1, a p, b Ž 1 p., te miimal value of Ž A,. uder te assumptio Ž A. p is equal to Ž 2.1. R Ž p. mi Ž Ž p.., 1 Ž Ž 1 p.. 4. We te measure is symmetric aroud its media, te expressio Ž 2.1. is simplified: Ž 2.2. R Ž p. Ž p..

4 38 S. BOBKOV By Propositio 2.2, property Ž a. i Teorem 1.1 is ecessary for extremality of te stadard alf-spaces Ž tis olds eve i te case 1.. Now we explai ow to prove ecessity of Ž b. ad Ž c.. Note tat te stadard alf-spaces A x : x Ž p.4 p 1 are ex- tremal i te isoperimetric problem Ž 1.1. if ad oly if r p. Ž p. ; Ž. tat is, r Ž p. R Ž p., for ay 0 p 1, 0 i te followig, R is defied by Ž Suppose tat te last idetity olds. Give p, q Ž 0, 1., cosider two alf-spaces A ad B x : x Ž q.4 p q 2 of -measure p ad q, respectively. Te te cube A p Bq as measure pq ad, terefore, sice Ž 1.1. is attaied at te alf-space A, we get Ž p q. Ž Ž p q A D A B D ; tat is Ž 2.4. R Ž pq. RŽ p. R Ž q. for all p, q Ž 0, 1., 0. Terefore, i order to prove te ecessity of Ž b. ad Ž c. i Teorem 1.1, it suffices to establis te followig lemma Ž see Sectio 4.. LEMMA 2.3. Property 2.4 ad a imply b ad c. To prove te sufficiecy part i Teorem 1.1, we will sow te followig i Sectio 5. LEMMA 2.4. Properties a ad c imply 2.4. We will also sow Ž Lemma 5.2. wy Ž c. may be replaced i tis lemma by te assumptio of log-cocavity of f. To complete te proof of Teorem 1.1, te problem ow is ow to derive te Ž. idetity r R from Ž 2.4. ad property Ž a.. or tis purpose, we use a statemet from ŽTeorems 1.1 ad 1.2, wic are formulated ere togeter for te space X wit te caoical elargemet.. Let be a probability Borel measure o. or ay 1, 0, defie te fuctio of p Ž 0, 1., p q Ž. if Ž A Ž, 1.., were te ifimum is over all Borel-measurable A of measure Ž A. p, ad were Ž, 1. is te ope cube i. Clearly, for absolutely cotiuous probability distributios, Ž. r Ž.. Ž1. THEOREM Suppose tat te fuctio is cocave ad icreasig Ž. o 0, 1. Te if is te maximal fuctio amog all icreasig Ž1. bijectios R i Ž 0, 1. suc tat, for ay p, q Ž 0, 1., Ž 2.5. RŽ pq. RŽ p. RŽ q., Ž 2.6. SŽ pq. SŽ p. SŽ q.,

5 EXTREMAL PROPERTIES 39 Ž1. were S p 1 R 1 p. I particular, if ad oly if te fuc- Ž1. tio R satisfies Ž 2.5. ad Ž I tis statemet, te parameter is fixed. irst let us ceck tat Teorem 2.5 may be applied to log-cocave mea- Ž1. sures satisfyig a ad b. By Propositio 2.2, R, ad tus we eed to sow tat R is cocave ad icreasig o Ž 0, 1.. Sice as a cotiuous positive desity f o te wole real lie, R icreases Ž strictly.. Sice logž f. is cocave, we ave tat, weever 0, te icremet logž fž x.. logž fž x.. logž fž x. fž x.. represets a oicreasig Ž fuctio o. Cosequetly, te derivative R p f Ž p.. Ž f Ž p.. does ot icrease, eiter. Tis implies cocavity of R Ž see also Propositio 6.1 about a coverse statemet wic sows tat Teorem 2.5 caot be applied to a more geeral class of probability distributios.. Tus, assumig Ž a. ad Ž b. i Teorem 1.1, oe ca apply Teorem 2.5 to. By Lemma 2.4, were Ž c. is also assumed, property Ž 2.4. is fulfilled; tat is, Ž olds for R R. It is obvious tat te fuctio SŽ p. 1 R Ž 1 p. is te iverse of R. Te Ž 2.6. may be writte as R Ž pq. R Ž p. R Ž q. for all p, q Ž 0, 1., wic is equivalet to Ž Cosequetly, Teorem 2.5 allows us to coclude tat Ž1. ; tat is, for all 1, Ž. Ž1., tat is, r Ž. R. Tus, te sufficiecy part i Teorem 1.1 will be proved. Now we sow tat if Ž 1.2. is fulfilled for M ad M max, , te te stadard alf-spaces are extremal for. Recall tat ad 2 are idepedet radom variables wit commo log-cocave law. Iequality Ž 1.2. wit M M says tat, for all 0, p Ž 0, 1., 1 Ž Ž. Ž Ž., a b,, were a p, b Ž 1 p.. Hece, by Propositio 2.1, te itervals Ž, a are extremal i te oe-dimesioal isoperimetric problem. By Propositio 2.2, is symmetric aroud a poit, ad tus property Ž a. i Teorem 1.1 is fulfilled. If Ž 1.2. is true for M, we obtai Ž for dimesio 2. Hece, Ž 2.4. is valid, too, ad by Lemma 2.3 properties Ž b. ad Ž c. i Teorem 1.1 are also fulfilled. Applyig tis teorem to te product of cocludes te argumet. 3. Oe-dimesioal isoperimetric problem for log-cocave measures. irst, we explai wy it suffices to prove Propositio 2.1 uder te followig additioal assumptio: Ž 3.1. fž x. 0 for all x Ž i.e., a, b., te derivative of logž f. exists everywere ad represets a cotiuous decreasig fuctio.

6 40 S. BOBKOV Ideed, te fuctio u logž f. is covex o Ž a, b. ; terefore, tere exists a sequece u of covex fuctios o suc tat te followig old: 1. Te derivative of u is a cotiuous icreasig fuctio; 2. u coverges poitwise to u o Ž a, b. ad to o a, b ; 3. f expž u. is te desity of some probability measure. Te f are log-cocave ad, moreover, satisfy Ž I additio, Ž A. Ž A., for all measurable A. I particular, Ž x. Ž x. for all x, p Ž p. for all p Ž 0, 1., were is te distributio fuctio of ad is te iverse of. Now, if oe may apply Propositio 2.1 to, te, accordig to Ž 2.1., for all measurable A, p Ž 0, 1., 0, we ave 4 Ž 3.2. Ž A D. mi Ž p., 1 Ž 1 p.. Takig te limit i 3.2, we obtai 4 Ž 3.3. Ž A D. mi Ž p., 1 Ž 1 p.. Ž Sice 3.3 provides a equality at te iterval A, Ž p. or at A Ž 1 p.,., Propositio 2.1 follows from Ž Now we may assume Ž or coveiece, it is better to cosider te exteded lie, ad subsets of it. or ay oempty set A, ad 0, we use te otatio A A,. Let A Ž a. p deote te iterval a, b, of -measure p, a b. Te variable a may vary i te iterval, Ž 1 p., ad b bž a. is te fuctio of a defied by te equality Ž 3.4. Ž b. Ž a. p. Ž. Note tat B a, a b,, were b b a, as measure 1 p. p LEMMA 3.1. or ay p Ž 0, 1., tere exists a a Ž p. suc tat Ž A Ž a p icreases i, a ad decreases i a, Ž 1 p Te same olds for Ž B Ž a.. ad some poit a Ž p.. p 0 Ž PROO. Clearly, te fuctio b icreases, b p, b Ž 1 p... Differetiatio of Ž 3.4. gives fž b. b fž a. 0. Differetiatig te fuctio Ž a. Ž A Ž a.. Ž b. Ž a., we obtai p f Ž a. Ž a. f Ž b. b f Ž a. f Ž b. f Ž a. f Ž b. f Ž b. f Ž a. f Ž a.. f Ž b. f Ž a. By assumptio Ž 3.1., te icremet už x. už x. of te fuctio už x. logž fž x.. represets a icreasig fuctio of x. Terefore, for eac

7 EXTREMAL PROPERTIES 41 0, te fuctio UŽ a. fž a. fž a. decreases ad VŽ a. fž b. fž b. icreases Ž as a compositio of two icreasig fuctios.. Tus, te fuctio Ž a. V Ž a. UŽ a. f Ž a. Ž is cotiuous ad icreasig o, Ž 1 p... Cosequetly, oe of te followig occurs: Ž a tere exists a poit a, Ž 1 p.. suc tat 0 o Ž, a. 0 0 Ž ad 0 o a, Ž 1 p.. 0 ; Ž b 0 o, Ž 1 p..; Ž c 0 o, Ž 1 p... I case Ž a., te first statemet of Lemma 3.1 olds wit a Ž p. 0 a 0; i cases b ad c, oe sould set a p Ž 1 p. 0 ad a0, respectively. Te proof of te secod part of Lemma 3.1 does ot differ from te above argumet. So, we ave proved Propositio 2.1 witi te class of all closed itervals a, b, of -measure greater ta or equal to p, ad te ext task is to exted tis propositio to disjoit uios A formed by closed itervals. Let Ip deote te family of suc sets A of -measure greater ta or equal to p; for eac A I, let rak Ž A. p deote te umber of itervals wic form A. I fact, because of approximatio argumet ad absolute cotiuity of, it is sufficiet to establis Propositio 2.1 for te class I oly. p LEMMA 3.2. or ay 0 p 1, Propositio 2.1 olds witi te family I. p PROO. Certaily, te parameter 0 is supposed to be arbitrary, too. Let I p deote te family of all sets A from Ip tat are formed by itervals a, b i i i, a1 b1 a b, wose closed -eigboroods a, b i i i do ot itersect. Now it suffices to prove te lemma witi Ip. Ideed, if i j for some i j, te i j a i, bj ad, terefore, two itervals i ad j ca be replaced by oe iterval a, b i j wic as a larger measure. Tus, for ay set A from I p, tere exists a set A from I p wit te same -eigborood. Let A I p be formed by te itervals i, 1 i, as above. We will move tese itervals, keepig teir measures p Ž. i i costat. I addi- tio, te iteriors of movig itervals sould ot itersect so tat te measure of teir uio remais costat, too Ž A.. Te rigt eds b bž a. i i sould deped o te appropriate left eds ai as i Lemma 3.1. Accordig to tis lemma, i te case a a Ž p. i 0 i, we may move ai to te left util ai bi if i 1, or ai if i 1. Respectively, i te case ai a Ž p. 0 i, we move ai to te rigt util bi ai if i, or bi if i. Applyig suc a procedure to ay of te middle itervals i, 1 i, 3, we obtai a ew iterval of measure p ad a ew set A i i 1 j,

8 42 S. BOBKOV for j i ad for j i Ž j j j i oly oe iterval i were 1 j, i A as bee caged.. Te Ž A. Ž A.. By Lemma 3.1, Ý Ý 1 ž / 1 j j Ž 3.5. A Ž A., sice A Ž. 1 j ad sice j are disjoit. By te defiitio of A 1, rak Ž A. 1 1; tat is, A1 cosists of 1 itervals. Oe ca cotiue tis process ad costruct A 2,..., A 2. Te last set will cosist of two itervals. Agai, applyig te above procedure to tese two itervals, we obtai a set of B of te followig tree types: B, a ; B b, ; B, a b,. I all te cases, Ž B. Ž A. ad Ž B. Ž A., sice Ž 3.5. was used at eac step. To exclude te sets of te tird type, it remais to apply te secod part of Lemma 3.1. REMARK 3.3. I order to prove Propositio 2.1, log-cocavity was used i Lemma 3.1. Neverteless, te statemet olds for some oter distributios, too. I particular, te reasoig i te proof of tis lemma ca be applied to a arbitrary distributio wic is cocetrated o a iterval were as a mootoic desity. PROO O PROPOSITION 2.2. Due to Propositio 2.1, tere is otig to prove for symmetric measures. Coversely, assume te itervals A of te form A Ž, a represet te sets for wic te value Ž A. is miimal provided Ž A. p. Cosequetly, for sets A Ž, a ad B b,. of -measure p Ž 0, 1., oe as Ž A. Ž B., for all 0; tat is, Ž 3.6. Ž Ž p.. 1 Ž Ž 1 p... or 0, Ž 3.6. turs ito a equality, ad we obtai te iequality for Ž. Ž derivatives of bot parts of 3.6 at 0: f p f Ž 1 p... Replacig p wit 1 p, we get a coverse iequality. Tus, for all p Ž 0, 1., Ž 3.7. f Ž Ž p.. f Ž Ž 1 p... Ž. Ž Te fuctio : 0, 1 a, b is smoot, ad p 1f Ž p... Ž. Ž So, by 3.7, p Ž 1 p.., for all p Ž 0, 1.. Terefore, for some costat m, p Ž 1 p. 2m, for all p Ž 0, 1.. However, tis meas tat m is te media of, ad is symmetric aroud m. Te proof is complete. 4. Necessity. PROO O LEMMA 2.3. Suppose tat, for all p 0, 1, 0, Ž 4.1. RŽ pq. R Ž p. RŽ q., Ž were R p Ž p... or 0, Ž 4.1. turs ito a equality, ad we obtai te iequality for derivatives of bot parts of Ž 4.1. at 0: f Ž Ž pq.. f Ž Ž p.. q f Ž Ž q.. p.

9 EXTREMAL PROPERTIES 43 Tis iequality coicides wit Ž 1.3., ad ecessity of Ž c. i Teorem 1.1 as bee proved. To prove Ž b., ote tat, for ay fuctio R: Ž 0, 1. Ž0, 1 satisfyig Ž 4.1., RŽ p. RŽ p., for all p Ž 0, 1.; terefore, RŽ p. 0, as p 0, if R 1. I te case R R, we ave RŽ p. Ž a., as p 0 recall tat a Ž 0.. If a, te, for b a, RŽ 0. 0; cosequetly R does ot satisfy Ž Tus, a is ecessary. Sice is symmetric aroud its media, we obtai b Ž 1., ad te lemma as bee establised. 5. Sufficiecy. We prove Lemma 2.4 witout assumptio about log-cocavity of measure. Let a distributio fuctio ave te cotiuous Ž. positive desity f. As usual, R p p, were is te iverse of. LEMMA 5.1. If, for ay p, q 0, 1, f Ž pq. f Ž p. f Ž q. Ž 5.1., pq p q te, for all p, q 0, 1, 0, Ž 5.2. R Ž pq. RŽ p. R Ž q.. PROO. Step 1. Assume tat iequality Ž 5.1. is strict for all p, q. Te family R,, forms a oe-parameter group of icreasig bijec- tios i Ž 0, 1.: for ay,, R is te compositio of R ad R, ad R is te iverse of R. So, if Ž 5.2. olds for ad, te it olds also for. ix p, q Ž 0, 1. ad set JŽ p, q. 0: Ž 5.2. wit p ad q is valid for all 0 4. We eed to sow tat JŽ p, q. 0,., for all p, q Ž 0, 1.. Te set JŽ p, q. is closed i 0,., sice R is a cotiuous fuctio of. Hece, it is sufficiet to sow tat if JŽ p, q., te JŽ p, q., for all small Ž eoug. Set p R p ad q R q, ad defie I t f Ž t.., 0 t 1. Te, for all t Ž 0, 1., R Ž t. t IŽ t. ož., as 0, by Taylor expasio. I particular, Ž 5.3. Ž. Ž. Ž. R Ž p. R R Ž p. p IŽ p. ož., R q R R q q I q o, R pq R R pq I R pq o. Te first two expasios give Ž 5.4. R Ž p. R Ž q. p q Ž p IŽ q. q IŽ p.. ož.. Sice JŽ p, q., we ave R Ž pq. R Ž p. R Ž q. p q. If R Ž pq. p q, te, from Ž 5.3. ad Ž 5.4., we obviously get R Ž pq. R Ž p. R Ž q., for all small eoug. If R Ž pq. p q, te te last iequality olds because

10 44 S. BOBKOV Ž. Ž I p q p I q. q IŽ p., wic follows from te strict variat of Ž I bot cases, we obtai JŽ p, q.. Step 2 Ž Geeral case.. Let Ž x. ŽTŽ x.. T, were T is a smoot covex fuctio from to suc tat te derivative T is icreasig ad positive o, T, T. Let T ad T be te iverse of T ad te iverse of T, respectively. We first sow tat T satisfies te assumptio of Step 1. Ideed, terefore, f Ž x. Ž x. f T Ž x. T Ž x., Ž p. T Ž p. ; T T T Ž. I Ž p. f Ž p. f Ž p. T T Ž p. IŽ p. Ž p., T T T Ž Ž were p T T Ž p... Te fuctio icreases i Ž 0, 1. as it is a compositio of icreasig fuctios; ece, for all p, q Ž 0, 1., Ž pq. mi Ž p., Ž q.4. Usig Ž 5.1. via te fuctio I, we ave, for all p, q Ž 0, 1., IT Ž pq. IŽ pq. IŽ p. IŽ q. Ž pq. Ž pq. Ž pq. pq pq p q IŽ p. IŽ q. IT Ž p. IT Ž q. Ž p. Ž q.. p q p q Tus, T satisfies te assumptio of Step 1, ad oe may coclude tat, for all p, q Ž 0, 1., 0, Ž 5.5. Ž pq. Ž p. Ž q.. T T T T T T Now, if TŽ x. T Ž x. x, as, poitwise, te Ž x. Ž x. T ad p Ž p., for all x, p Ž 0, 1. T. or example, oe may set T Ž x. x x logž 1 x.. Cosequetly, 5.2 follows from 5.5 by takig te limit as, ad Lemma 5.1 is proved. As we ave already see Ž Lemma 2.3., Ž 5.1. is also ecessary for Ž Now we preset oe sufficiet Ž but ot ecessary. coditio wic allows us to coclude Ž LEMMA 5.2. If f is a cotiuous ad positive desity of o te real lie suc tat te fuctio logž f. is cocave, te Ž 5.2. is fulfilled for all 0 ad p, q Ž 0, 1.. PROO. Te fuctio Ž. Ž. Ž x. log R expž x. log expž x. is positive, cotiuous ad icreasig o Ž 0,., Ž 0. 0 ad Ž 5.2. meas tat, for all x, y 0, 5.6 x y x y.

11 EXTREMAL PROPERTIES 45 It is well kow tat every icreasig, positive ad covex fuctio o Ž 0,. suc tat Ž 0. 0 satisfies Ž Hece, to coclude Ž 5.2., it suffices to sow tat is covex. Note tat is differetiable, ad its derivative R Ž expž x.. Ž x. expž x. R expž x. Ž Ž.. f Ž Ž expž x... Ž Ž expž x... f exp x exp x does ot decrease o 0, if ad oly if te fuctio f Ž p. p f Ž p. Ž p. does ot icrease o Ž 0, 1., or usig te cage of variable p Ž x. te fuctio Ž 5.7. f Ž x. Ž x. f Ž x. Ž x. does ot icrease o. Tis is true, sice, for ay 0, Ž 5.7. is te icremet of logž f. o te iterval x, x. Te proof is complete. APPENDIX A probability Borel measure o te locally covex space E is called log-cocave if, for all oempty Borel sets A, B E ad 0, 1, 1 Ž A Ž 1. B. Ž A. Ž B., were A Ž 1. B a Ž 1. b: a A, b B 4, ad were deotes te ier measure. A full descriptio of log-cocave measures was give by Borell 3. I te case E, ad if is ot a uit mass x at some poit x, te above defiitio is reduced to tat metioed i Sectio 1: as a desity f suc tat te fuctio logž f. is cocave o Žas a fuctio wit values i,... I tis sectio, we give oter equivalet defiitios of log-cocavity for measures o te real lie. Let be a oatomic probability measure wit Ž cotiuous. distributio fuctio Ž x. ŽŽ, x., x. Set a if x : Ž x. 04, b sup x : Ž x. 14. Assume tat strictly icreases o a, b, ad let : Ž a, b. Ž 0, 1. deote te iverse of restricted to Ž a, b.. PROPOSITION A.1. are equivalet: Uder te above assumptios, te followig properties Ž a. is log-cocave; Ž b for all 0, te fuctio R p Ž p.. is cocave o Ž 0, 1.;

12 46 S. BOBKOV Ž c. as a cotiuous, positive desity f o Ž a, b., ad, moreover, te Ž fuctio I p f Ž p.. is cocave o Ž 0, 1.. PROO. Ž a. Ž b. Tis was sow i Sectio 2. Ž a. Ž c. By assumptio, as a positive, cotiuous desity f o Ž a, b. suc tat logž f. is cocave. Hece, f is absolutely cotiuous, ad, moreover, tere exists a RadoNikodym derivative f for f o Ž a, b. suc tat te fuctio f f is a oicreasig RadoNikodym derivative of logž f. o Ž a, b.. Te sice is cotiuously differetiable, te fuctio I f Ž p. Ž p. f Ž p. represets a RadoNikodym derivative of I o Ž 0, 1.. Clearly, I does ot icrease; ece, I is cocave. Ž c. Ž a. By assumptio, I is positive ad cocave o Ž 0, 1.; ece, tere exists a oicreasig RadoNikodym derivative I. Sice is cotiuously differetiable o Ž a, b., te fuctio I ŽŽ x.. fž x. represets a Rado Nikodym derivative of IŽŽ x... However, IŽŽ x.. fž x., for all x Ž a, b.; ece, f is absolutely cotiuous ad, moreover, f I ŽŽ x.. fž x. is a RadoNikodym derivative of f x. Terefore, I Ž. f f represets a RadoNikodym derivative of logž f.. Sice I Ž. does ot icrease, logž f. is cocave o Ž a, b.. Ž b. Ž a. or simplicity, we assume a, b Žmior cages sould be made i cases a ador b.. or ay 0, R is cocave; ece, tere exists a oicreasig RadoNikodym derivative L of R wic ca also be cose to be cotiuous from te rigt. We also ave L Ž p. 0, for all p Ž 0, 1. sice, i additio to cocavity, R icreases ad R Ž p. 1, as p 1. Usig R ŽŽ x.. Ž x. oe ca write Ž y. Ž y. Ž x. L Ž t. dt, H Ž x. wic olds for all x, y ad 0. Lettig y x, we get te followig: Ž y. Ž x. LŽ Ž x.. Ž Ž y. Ž x.. ož Ž y. Ž x.., y x ; Ž y. Ž x. L Ž x. Ž y. Ž x. o Ž y. Ž x., y x. I particular, if is differetiable at x, te tere exist limits Ž y. Ž x. fr Ž x. lim LŽ Ž x.. Ž x., yx y x Ž y. Ž x. fl Ž x. lim LŽ Ž x.. Ž x.. yx y x Sice 0 is arbitrary ad sice, by te Lebesgue teorem, is differetiable at almost all x, te value of x ca be arbitrary; so we coclude

13 EXTREMAL PROPERTIES 47 tat te values f Ž x. ad f Ž x. r l are well defied for all x ad tat fr ad fl represet te rigt ad left derivatives of o te wole real lie. I additio, tese fuctios satisfy Ž A.1. fr Ž x. LŽ Ž x.. fr Ž x., Ž. A.2 f x L x f x, l l for every x ad 0. Usig Ž A.2., ote tat if f Ž x. l 0 0 at some poit x te f 0 everywere o x,. 0 l 0. However, fr fl almost everywere; ece, we would ave f Ž x. 0 at some poit x x ad, by Ž A.1. l 1 1 0, f 0 everywere o x,. r 1. As a result, 0; tat is, is costat o x,.. Te last is impossible; cosequetly, f Ž x. 1 l 0, for all x. By te same argumet, for all x, f Ž x. r 0. Now, oe may itroduce te fuctios g logž f. ad g logž f. r r l l ad may rewrite Ž A.1. ad Ž A.2. as g Ž x. g Ž x. L Ž x., g Ž x. g Ž x. L Ž x.. r r l l We observe tat icremets of gr ad gl represet oicreasig fuctios. Terefore, gr ad gl are cocave, if tey are cotiuous. or every x, y, x y, ad every 0, we ave Ž A.3. gr Ž y. gr Ž x. gr Ž y. gr Ž x.. ixig x ad lettig y x i A.3, we get Ž A.4. g Ž x. g Ž x., ad fixig y ad lettig x y i A.3, we get Ž A.5. g Ž y. g Ž y.. r r By Ž A.4., if g Ž x. 0 at some poit x, te g Ž x. r 0 0 r 0, for all x x 0; ad, by Ž A.5. if g Ž y. 0 at some poit y, te g Ž y. r 0 0 r 0, for all y y 0. Hece, if gr is discotiuous at a poit x, te gr is discotiuous at eac poit i Ž x,.. However, gr is cotiuous at every poit from a set of te secod category. Ideed, te fuctio f is a poitwise limit 1 fr Ž x. lim ž x Ž x. r r r ž / / of te sequece of cotiuous fuctios, ad te same cocers te fuctio g. Hece, g belogs to te first class of Baire, ad, by Baire s teorem Ž r r see, e.g., 5, Sectio 31., all te poits were gr is discotiuous form a set of te first category. Tus, for all x, g Ž x. g Ž x. 0; tat is, g ad Ž 0 r 0 r 0 r by te same argumet. gl are cotiuous o te wole real lie. Cosequetly, tese fuctios are cocave. We also obtai tat te fuctios fr ad fl are cotiuous, ad sice f f almost everywere, we coclude tat f Ž x. f Ž x. Ž x. r l r l, for all x. Terefore, te derivative f represets a positive desity of suc tat te fuctio logž f. is cocave. Te proof is complete.

14 48 S. BOBKOV Ackowledgmets. Te autor is grateful to te Departmet of Statistics of te Uiversity of Nort Carolia at Capel Hill for te ivitatio to visit Te Ceter for Stocastic Processes. Te autor taks te referee for careful readig ad useful commets. REERENCES BOBKOV, S. G. Ž Isoperimetric problem for uiform elargemet. Tecical Report 394, Dept. Statistics, Uiv. Nort Carolia. 2 BOBKOV, S. G. Ž Isoperimetric iequalities for distributios of expoetial type. A. Probab BORELL, C. Ž Covex measures o locally covex spaces. Ark. Mat BORELL, C. Ž Te BruMikowski iequality i Gauss space. Ivet. Mat KURATOWSKI, A. Ž Topology 1. Academic Press, New York. 6 SUDAKOV, V. N. ad TSIREL SON, B. S. Ž Extremal properties of alf-spaces for sperically ivariat measures. J. Soviet Mat Traslated from Zap. Nauc. Sem. Leigrad. Otdel. Mat. Ist. Steklov. 41 Ž Ž i Russia.. 7 TALAGRAND, M. Ž A ew isoperimetric iequality ad te cocetratio of measure peomeo. Israel Semiar Ž GAA.. Lecture Notes i Mat Spriger, Berli. DEPARTMENT O MATHEMATICS SYKTYVKAR UNIVERSITY SYKTYVKAR RUSSIA root@sucit.komi.su