Mixed f-divergence for multiple pairs of measures

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1 Mxed f-dvergece for multple pars of measures Elsabeth M. Werer ad Depg Ye Abstract I ths paper, the cocept of the classcal f-dvergece for a par of measures s exteded to the mxed f-dvergece for multple pars of measures. The mxed f-dvergece provdes a way to measure the dfferece betwee multple pars of probablty measures. Propertes for the mxed f-dvergece are establshed, such as permutato varace ad symmetry dstrbutos. A Alexadrov-Fechel type equalty ad a sopermetrc equalty for the mxed f-dvergece are proved. 1 Itroducto I applcatos such as patter matchg, mage aalyss, statstcal learg, ad formato theory, oe ofte eeds to compare two probablty measures ad eeds to kow whether they are smlar to each other. Hece, fdg the rght quatty to measure the dfferece betwee two probablty measures P ad Q s cetral. Tradtoally, people use the classcal L p dstaces betwee P ad Q, such as the varatoal dstace ad the L 2 dstace. However, the famly of f-dvergeces s ofte more sutable to fulfll the goal tha the classcal L p dstace of measures. The f-dvergece D f P, Q of two probablty measures P ad Q was frst troduced 8] ad depedetly 2, 30] ad was defed by p D f P, Q f q dµ. 1.1 q Here, p ad q are desty fuctos of P ad Q wth respect to a measure µ o. The dea behd the f-dvergece s to replace, for stace, the fucto ft t 1 the varatoal dstace by a geeral covex fucto f. Hece the f-dvergece cludes varous wdely used dvergeces as specal cases, such as, the varatoal dstace, the Kullback-Lebler dvergece 16], the Bhattacharyya dstace 5] ad may more. Cosequetly, the f-dvergece receves cosderable atteto ot oly the formato theory e.g., 3, 7, 14, 17, 31] but also may other areas. We oly meto covex geometry. Wth the last few years, amazg coectos have bee dscovered betwee otos ad cocepts from covex geometry ad formato theory, e.g., 9,10,15,24,25,32], leadg to a totally ew Keywords: Alexadrov-Fechel equalty, f-dssmlarty, f-dvergece, sopermetrc equalty. Mathematcs Subject Classfcato Number: 28, 52, 60. Partally supported by a NSF grat Supported by a NSERC grat ad a start-up grat from Memoral Uversty of Newfoudlad 1

2 pot of vew ad troducg a whole ew set of tools the area of covex geometry. I partcular, t was observed 38] that oe of the most mportat affe varat otos, the L p -affe surface area for covex bodes, e.g., 18 20, 22, 34], s Réy etropy from formato theory ad statstcs. Réy etropes are specal cases of f-dvergeces ad cosequetly those were the troduced for covex bodes ad ther correspodg etropy equaltes have bee establshed 39]. We also refer to, for stace 4], for more refereces related to the f-dvergece. Exteso of the f-dvergece from two probablty measures to multple probablty measures s fudametal may applcatos, such as statstcal hypothess test ad classfcato, ad much research has bee devoted to that, for stace 28,29,42]. Such extesos clude, e.g., the Matusta s affty 26,27], the Toussat s affty 37], the formato radus 36] ad the average dvergece 35]. The f-dssmlarty D f P 1,, P l for probablty measures P 1,, P l, troduced 11,12] for a covex fucto f : R l R, s a atural geeralzato of the f-dvergece. It s defed as D f P 1,, P l fp 1,, p l dµ, where the p s are desty fuctos of the P s that are absolutely cotuous wth respect to µ. For a covex fucto f, the fucto fx, y yf x y s also covex o x, y > 0, ad D f P, Q s equal to the classcal f-dvergece defed formula 1.1. Note that the Matusta s affty s related to fx 1,, x l l x 1/l, ad the Toussat s affty s related to fx 1,, x l l l a 1. xa, where a 0 ad such that Here, we troduce specal f-dssmlartes, amely the mxed f-dvergece ad the -th mxed f- dvergece, whch ca be vewed as vector forms of the usual f-dvergece. We establsh some basc propertes of these quattes, such as permutato varace ad symmetry dstrbutos. We prove a sopermetrc type equalty ad a Alexadrov-Fechel type equalty for the mxed f-dvergece. Alexadrov-Fechel equalty s a fudametal equalty covex geometry ad may mportat equaltes such as the Bru-Mkowsk equalty ad Mkowsk s frst equalty follow from t see, e.g., 9, 33]. The paper s orgazed as follows. I Secto 2 we establsh some basc propertes of the mxed f-dvergece, such as permutato varace ad symmetry dstrbutos. I Secto 3 we prove the geeral Alexadrov-Fechel equalty ad sopermetrc equalty for the mxed f-dvergece. Secto 4 s dedcated to the -th mxed f-dvergece ad ts related sopermetrc type equaltes. 2 The Mxed f-dvergece Throughout ths paper, let, µ be a fte measure space. For 1, let P p µ ad Q q µ be probablty measures o that are absolutely cotuous wth respect to the measure µ. Moreover, we assume that for all 1,,, p ad q are ozero µ-a.e. We use P ad Q to deote the vectors of probablty measures, or, short, probablty vectors, P P 1, P 2,, P, Q Q1, Q 2,, Q. 2

3 We use p ad q to deote the vectors of desty fuctos, or desty vectors, for P ad Q respectvely, d P dµ p p 1, p 2,, p, d Q dµ q,,, q. We make the coveto that 0 0. Deote by R + {x R : x 0}. Let f : 0, R + be a o-egatve covex or cocave fucto. The -adjot fucto f : 0, R + of f s defed by f t tf1/t. It s obvous that f f ad that f s aga covex, respectvely cocave, f f s covex, respectvely cocave. Let f : 0, R +, 1, be ether covex or cocave fuctos. Deote by f f 1, f 2,, f the vector of fuctos. We wrte to be the -adjot vector for f. f f 1, f 2,, f Now we troduce the mxed f-dvergece for f, P, Q as follows. Defto 2.1. Let, µ be a measure space. Let P ad Q be two probablty vectors o wth desty vectors p ad q respectvely. The mxed f-dvergece D f P, Q for f, P, Q s defed by D f P, Q Smlarly, we defe the mxed f-dvergece for f, Q, P by D f Q, P p f q dµ. 2.2 q q f p dµ. 2.3 p A specal case s whe all dstrbutos P ad Q are detcal ad equal to a probablty dstrbuto P. I ths case, D f P, Q D f1,f 2,,f P, P,, P, P, P,, P f 1] 1. Let π S deote a permutato o {1, 2,, } ad deote π p p π1, p π2,, p π. Oe mmedate result from Defto 2.1 s the followg permutato varace for D f P, Q. 3

4 Proposto 2.1 Permutato varace. Let the vectors f, P, Q be as above, ad let π S be a permutato o {1, 2,, }. The D f P, Q D π f π P, π Q. Whe all f, P, Q are equal to f, P, Q, the mxed f-dvergece s equal to the classcal f- dvergece, deoted by D f P, Q, whch takes the form p D f P, Q D f,f,,f P, P,, P, Q, Q,, Q f qdµ. q As f t tf1/t, oe easly obtas a fudametal property for the classcal f-dvergece D f P, Q, amely, D f P, Q D f Q, P, for all f, P, Q. Smlar results hold true for the mxed f-dvergece. We show ths ow. Let 0 k. We wrte D f,k P, Q for D f,k P, Q k p f q q k+1 Clearly, D f, P, Q D f P, Q ad D f,0 P, Q D f Q, P, where f f 1, f 2,, f. f q The we have the followg result for chagg order of dstrbutos. p p ] 1 dµ. Proposto 2.2 Prcple for chagg order of dstrbutos. Let f, P, Q be as above. The, for ay 0 k, oe has D f P, Q D f,k P, Q. I partcular, Proof. Let 0 k. The, D f P, Q D f P, Q D f Q, P. k k D f,k P, Q, p f q q p f q q where the secod equalty follows from f p q q f k+1 k+1 q p p. p f q dµ q f q p p ] 1 dµ A drect cosequece of Proposto 2.2 s the followg symmetry prcple for the mxed f- dvergece. 4

5 Proposto 2.3 Symmetry dstrbutos. Let f, P, Q be as above. The, D f P, Q+D f P, Q s symmetrc P ad Q, amely, D f P, Q + D f P, Q D f Q, P + D f Q, P. Remark. Proposto 2.2 says that D f P, Q remas the same f oe replaces ay trple f, P, Q by f, Q, P. It s also easy to see that, for all 0 k, l, oe has Hece, for all 0 k, l, s symmetrc P ad Q. D f P, Q D f,k P, Q D f,l Q, P D f Q, P. D f,k P, Q + D f,l P, Q D f P, Q + D f P, Q Hereafter, we oly cosder the mxed f-dvergece D f P, Q defed formula 2.2. Propertes for the mxed f-dvergece D f Q, P defed 2.3 follow alog the same les. Now we lst some mportat mxed f-dvergeces. Examples. The total varato s a wdely used f-dvergece to measure the dfferece betwee two probablty measures P ad Q o, µ. It s related to fucto ft t 1. Smlarly, the mxed total varato s defed by D T V P, Q p q 1 dµ. It measures the dfferece betwee two probablty vectors P ad Q. For a R, we deote by a + max{a, 0}. The mxed relatve etropy or mxed Kullback Lebler dvergece of P ad Q s defed by D KL P, Q Df+,,f + P, Q p l where ft t l t. Whe P P pµ ad Q Q qµ for all 1, 2,,, we get the followg modfed relatve etropy or Kullback Lebler dvergece ] q D KL P Q p l dµ. p + For the covex ad/or cocave fuctos f α t t α, α R for 1, the mxed Hellger tegrals s defed by D fα1,f α2,,f α P, Q 5 p α q 1 α q p ] dµ. ] 1 + dµ,

6 I partcular, D t α,t α,,t α P, Q p α α dµ. Those tegrals are related to the Toussat s affty 37], ad ca be used to defe the mxed α-réy dvergece D α {P Q } 1 α 1 l p α α dµ 1 α 1 l ] D t α,t α,,t α P, Q. The case α 1 2, for all 1, 2,,, gves the mxed Bhattacharyya coeffcet or mxed Bhattacharyya dstace of P, Q, D t, t,, t P, Q p Ths tegral s related to the Matusta s affty 26, 27]. For more formato o the correspodg f-dvergeces we refer to e.g. 17]. v I vew of exstg coectos betwee formato theory ad covex geometry e.g., 32, 38, 39], we defe the mxed f-dvergeces for covex bodes covex ad compact subsets R wth oempty terors K wth postve curvature fuctos f K, 1, s va the measures dµ. dp K 1 h K dσ ad dq K f K h K dσ, 1. Here, σ s the sphercal measure of the ut sphere S 1, h K u max x K x, u s the support fucto of K, ad f K u s the curvature fucto of K at u S 1, the recprocal of the Gauss curvature at x o the boudary of K wth ut outer ormal u. If f : 0, R +, 1, are covex ad/or cocave fuctos, the D f PK1,..., P K, Q K1,..., Q K S 1 f 1 f K h +1 K f K h K ] 1 dσ, are the geeral mxed affe surface areas troduced 41]. We refer to 33] for more detals o covex bodes. 3 Iequaltes The classcal Alexadrov-Fechel equalty for mxed volumes of covex bodes s a fudametal result covex geometry. A geeral verso of ths equalty for mxed volumes of covex bodes ca be foud 1, 6, 33]. Alexadrov-Fechel type equaltes for mxed affe surface areas ca be foud 21, 22, 40, 41]. Now we prove a equalty for the mxed f-dvergece for measures, whch we call 6

7 a Alexadrov-Fechel type equalty because of ts formal resemblace to be a Alexadrov-Fechel type equalty for covex bodes. Followg 13], we say that two fuctos f ad g are effectvely proportoal f there are costats a ad b, ot both zero, such that af bg. Fuctos f 1,..., f m are effectvely proportoal f every par f, f j, 1, j m s effectvely proportoal. A ull fucto s effectvely proportoal to ay fucto. These otos wll be used the ext theorems. For a measure space, µ ad probablty destes p ad q, 1, we put ad for j 0,, m 1, g 0 u g j+1 u m For a vector p, we deote by p,k the followg vector p f q, 3.4 q p j f j q j. 3.5 q j p,k p 1,, p m, p k,, p }{{ k, k > m. } m Theorem 3.1. Let, µ be a measure space. For 1, let P ad Q be probablty measures o, µ wth desty fuctos p ad q respectvely µ-a.e. Let f : 0, R +, 1, be covex fuctos. The, for 1 m, D f P, Q ] m k m+1 D f,k P,k, Q,k. Equalty holds f ad oly f oe of the fuctos g 1 m 0 g, 1 m, s ull or all are effectvely proportoal µ-a.e. If m, D f P, Q] D f P, Q, wth equalty f ad oly f oe of the fuctos f j pj q j q j, 0 j, s ull or all are effectvely proportoal µ-a.e. Remarks. I partcular, equalty holds Theorem 3.1 f all P, Q cocde, ad f λ f for some covex postve fucto f ad λ 0, 1, 2,,. Theorem 3.1 stll holds true f the fuctos f are cocave. 7

8 Proof. We let g 0 ad g j+1, j 0,, m 1 as 3.4 ad 3.5. By Hölder s equalty see 13] D f P, Q] m m 1 j0 m g 0 ug 1 u g m u dµ m 1 j0 k m+1 g 0 ug j+1 u m ] 1 m dµ m g 0 ugj+1u m dµ D f,k P,k, Q,k. Equalty holds Hölder s equalty, f ad oly f oe of the fuctos g 1 m 0 g, 1 m, s ull or all are effectvely proportoal µ-a.e. I partcular, ths s the case, f for all 1,,, P, Q P, Q ad f λ f for some covex fucto f ad λ 0. We requre some propertes of f-dvergeces for our ext result. Let f : 0, R + be a covex fucto. By Jese s equalty, p D f P, Q f q dµ f p dµ f1, 3.6 q for all pars of probablty measures P, Q o, µ wth ozero desty fuctos p ad q respectvely µ-a.e. Whe f s lear, equalty holds trvally 3.6. Whe f s strctly covex, equalty holds true f ad oly f p q µ-a.e. If f s a cocave fucto, Jese s equalty mples D f P, Q p f q dµ f q p dµ f1, 3.7 for all pars of probablty measures P, Q. Aga, whe f s lear, equalty holds trvally. Whe f s strctly cocave, equalty holds true f ad oly f p q µ-a.e. For the mxed f-dvergece wth cocave fuctos, oe has the followg result. Theorem 3.2. Let, µ be a measure space. For all 1, let P ad Q be probablty measures o whose desty fuctos p ad q are ozero µ-a.e. Let f : 0, R +, 1, be cocave fuctos. The D f P, Q] D f P, Q f If addto, all f are strctly cocave, equalty holds f ad oly f there s a probablty desty p such that for all 1, 2,, p q p, µ a.e. 8

9 Proof. Theorem 3.1 ad the remark after mply that for all cocave fuctos f, D f P, Q] D f P, Q f 1, where the secod equalty follows from equalty 3.7 ad f 0. Suppose ow that for all, p q p, µ-a.e., where p s a fxed probablty desty. The equalty holds trvally 3.8. Coversely, suppose that equalty holds 3.8. The, partcular, equalty holds Jese s equalty whch, as oted above, happes f ad oly f p q for all. Thus, D f P, Q 1] f 1/ / 1... q 1/ dµ. Note also that f all f : 0, R + are strctly cocave, f 1 0 for all 1. Equalty characterzato Hölder s equalty mples that all q are effectvely proportoal µ-a.e. As all q are probablty measures, they are all equal µ-a.e. to a probablty measure wth desty fucto say p. Remark. If f t a t + b are all lear ad postve, the equalty holds f ad oly f all p, q are equal µ-a.e. as covex combatos,.e., f ad oly f for all, j a a + b p + b a + b q a j a j + b j p j + b j a j + b j q j, µ a.e. 4 The -th mxed f-dvergece Let, µ be a measure space. Throughout ths secto, we assume that the fuctos f 1, f 2 : 0, {x R : x > 0}, are covex or cocave, ad that P 1, P 2, Q 1, Q 2 are probablty measures o wth desty fuctos p 1, p 2,, whch are ozero µ-a.e. We also wrte f f 1, f 2, P P1, P 2, Q Q1, Q 2. Defto 4.1. Let R. The -th mxed f-dvergece for f, P, Q, deoted by D f P, Q;, s defed as ] ] D f P, Q; p1 p2 f 1 f 2 dµ. 4.9 Remarks. Note that the -th mxed f-dvergece s defed for ay combato of covexty ad cocavty of f 1 ad f 2, amely, both f 1 ad f 2 cocave, or both f 1 ad f 2 covex, or oe s covex the other s cocave. It s easly checked that D f P, Q; D f2,f 1 P2, Q 2, P 1, Q 1 ;. 9

10 If 0 s a teger, the the trple f 1, P 1, Q 1 appears -tmes whle the trple f 2, P 2, Q 2 appears tmes D f P, Q;. Note that f 0, the D f P, Q; D f2 P 2, Q 2, ad f the D f P, Q; D f1 P 1, Q 1. Aother specal case s whe P 2 Q 2 µ almost everywhere ad µ s also a probablty measure. The such a -th mxed f-dvergece, deoted by D f 1, P 1, Q 1, ; f 2, has the form D f 1, P 1, Q 1, ; f 2 f2 1] 1 / ] p1 f 1 dµ. Examples ad Applcatos. For ft t 1, we get the -th mxed total varato D T V P, Q; p 1 p2 dµ. For f 1 t f 2 t t l t] +, we get the modfed -th mxed relatve etropy or -th mxed Kullback Lebler dvergece D KL P, Q; p 1 l p1 ] + p 2 l p2 ] For the covex or cocave fuctos f αj t t α j, j 1, 2, we get the -th mxed Hellger tegrals I partcular, for α j α, for j 1, 2, + dµ. D fα1,f α2 P, Q; p α 1 1 q1 α 1 1 p α 2 2 q1 α 2 2 dµ. D fα,f P α, Q; p α 1 q1 1 α p α 2 α 2 Ths tegral ca be used to defe the -th mxed α-réy dvergece D α P, Q; 1 α 1 l D fα,f P α, Q; ]. The case α 1 2 for all gves D t, t P, Q; p 1 2 p2 2 dµ, the -th mxed Bhattacharyya coeffcet or -th mxed Bhattacharyya dstace of p ad q. v Importat applcatos are aga the theory of covex bodes. As secto 2, let K 1 ad K 2 be covex bodes wth postve curvature fucto. For l 1, 2, let dp Kl 1 h K l dσ ad dq Kl f Kl h Kl dσ. 10 dµ.

11 Let f l : 0, R, l 1, 2, be postve covex fuctos. The, we defe the -th mxed f-dvergece for covex bodes K 1 ad K 2 by D f PK1, P K2, Q K1, Q K2 ; S 1 f 1 1 f K1 h +1 K 1 f K1 h K1 ] f 2 These are the geeral -th mxed affe surface areas troduced 41]. 1 f K2 h +1 K 2 f K2 h K2 ] The followg result holds for all possble combatos of covexty ad cocavty of f 1 ad f 2. Proposto 4.1. Let f, P, Q be as above. If j k or k j, the D f P, Q; D f P, Q; j ] k k j D f P, Q; k ] j k j. Equalty holds trvally f k or j. Otherwse, equalty holds f ad oly f oe of the fuctos f p q q, 1, 2, s ull, or f p1 1 ad f p2 2 are effectvely proportoal µ-a.e. I partcular, ths holds f P 1, Q 1 P 2, Q 2 ad f 1 λf 2 for some λ > 0. dσ. Proof. By formula 4.9, oe has D f P, Q; ] p1 f 1 f 2 p2 ] dµ { ] j ] j } k k j p1 p2 f 1 f 2 { ] k ] k } j k j p1 p2 f 1 f 2 dµ D f P, Q; j ] k k j D f P, Q; k ] j k j, where the last equalty follows from Hölder s equalty ad formula 4.9. The equalty characterzato follows from the oe Hölder equalty. I partcular, f P 1, Q 1 P 2, Q 2, ad f 1 λf 2 for some λ > 0, equalty holds. Corollary 4.1. Let f 1 ad f 2 be postve, cocave fuctos o 0,. The for all P, Q ad for all 0, D f P, Q; ] f1 1] f 2 1]. If addto, f 1 ad f 2 are strctly cocave, equalty holds ff p 1 p 2 µ-a.e. Proof. Let j 0 ad k Proposto 4.1. The for all 0, D f P, Q; ] Df1 P 1, Q 1 ] D f2 P 2, Q 2 ] f 1 1] f 2 1], 11

12 where the last equalty follows from equalty 3.7. To have equalty, the above equaltes should be equaltes. Proposto 4.1 mples that f 1 p1 ad f 2 p2 are effectvely proportoal µ-a.e. As both f 1 ad f 2 are strctly cocave, Jese s equalty requres that p 1 ad p 2 µ-a.e. Therefore, equalty holds f ad oly f f 1 1 ad f 2 1 are effectvely proportoal µ-a.e. As both f 1 1 ad f 2 1 are ot zero, equalty holds ff p 1 p 2 µ-a.e. Remark. If f 1 t a 1 t + b 1 ad f 2 t a 2 t + b 2 are both lear, equalty holds Corollary 4.1 f ad oly f p, q, 1, 2, are equal as covex combatos,.e., a 1 a 1 + b 1 p 1 + b 1 a 1 + b 1 a 2 a 2 + b 2 p 2 + b 2 a 2 + b 2, µ a.e. Ths proof ca be used to establsh the followg result for D f 1, P 1, Q 1, ; f 2. Corollary 4.2. Let, µ be a probablty space. Let f 1 be a postve cocave fucto o 0,. The for all P 1, Q 1, for all cocave or covex postve fuctos f 2, ad for all 0, D f1, P 1, Q 1, ; f 2 ] f1 1] f 2 1]. If f 1 s strctly cocave, equalty holds f ad oly f P 1 Q 1 µ. Whe f 1 t at + b s lear, equalty holds f ad oly f ap 1 + b a + b µ-a.e. Corollary 4.3. Let f 1 be a postve covex fucto ad f 2 be a postve cocave fucto o 0,. The, for all P, Q, ad for all k, D f P, Q; k ] f1 1] k f 2 1] k. If addto, f 1 s strctly covex ad f 2 s strctly cocave, equalty holds f ad oly f p 1 p 2 µ-a.e. Proof. O the rght had sde of Proposto 4.1, let ad j 0. Let k. The D f P, Q; k ] Df1 P 1, Q 1 ] k D f2 P 2, Q 2 ] k f 1 1] k f 2 1] k. Here, the last equalty follows from equaltes 3.6, 3.7 ad k. To have equalty, the above equaltes should be equaltes. Proposto 4.1 mples that f p1 1 ad f p2 2 are effectvely proportoal µ-a.e. As f 1 s strctly covex ad f 2 s strctly cocave, Jese s equalty mples that p 1 ad p 2 µ-a.e. Therefore, as both f 1 1 ad f 2 1 are ot zero, equalty holds f ad oly f p 1 p 2 µ-a.e. Remark. If f 1 t a 1 t + b 1 ad f 2 t a 2 t + b 2 are both lear, equalty holds Corollary 4.3 f ad oly f p, q, 1, 2, are equal µ-a.e. as covex combatos,.e., a 1 a 1 + b 1 p 1 + b 1 a 1 + b 1 a 2 a 2 + b 2 p 2 + b 2 a 2 + b 2, µ a.e. Ths proof ca be used to establsh the followg result for D f 1, P 1, Q 1, k; f 2. 12

13 Corollary 4.4. Let, µ be a probablty space. Let f 1 be a postve covex fucto o 0,. The for all P 1, Q 1, for all postve cocave or covex fuctos f 2, ad for all k, D f1, P 1, Q 1, k; f 2 ] f1 1] k f 2 1] k. If f 1 s strctly covex, equalty holds f ad oly f P 1 Q 1 µ. Whe f 1 t at + b s lear, equalty holds f ad oly f ap 1 + b a + b µ-a.e. Corollary 4.5. Let f 1 be a postve cocave fucto ad f 2 be a postve covex fucto o 0,. The for all P, Q, ad for all k 0, D f P, Q; k ] f1 1] k f 2 1] k. If addto, f 1 s strctly cocave ad f 2 s strctly covex, equalty holds ff p 1 p 2 µ-a.e. Proof. Let 0 ad j Proposto 4.1. The D f P, Q; k ] D f1 P 1, Q 1 ] k D f2 P 2, Q 2 ] k f 1 1] k f 2 1] k. Here, the last equalty follows from equaltes 3.6, 3.7, ad k 0. To have equalty, the above equaltes should be equaltes. Proposto 4.1 mples that f 1 p1 ad f 2 p2 are effectvely proportoal µ-a.e. As f 1 s strctly cocave ad f 2 s strctly covex, Jese s equalty requres that p 1 ad p 2. Therefore, equalty holds f ad oly f f 1 1 ad f 2 1 are effectvely proportoal µ-a.e. As both f 1 1 ad f 2 1 are ot zero, equalty holds f ad oly f p 1 p 2 µ-a.e. Ths proof ca be used to establsh the followg result for D f 1, P 1, Q 1, k; f 2. Corollary 4.6. Let f 1 be a cocave fucto o 0,. The for all P 1, Q 1, for all cocave or covex fuctos f 2, ad for all k 0, D f1, P 1, Q 1, k; f 2 ] f1 1] k f 2 1] k. If f 1 s strctly cocave, equalty holds f ad oly f P 1 Q 1 µ. Whe f 1 t at + b s lear, equalty holds f ad oly f ap 1 + b a + b µ-a.e. Refereces 1] A.D. Aleksadrov, O the theory of mxed volumes of covex bodes. II. New equaltes betwee mxed volumes ad ther applcatos, Mat. Sb. N. S Russa] 2] M.S. Al ad D. Slvey, A geeral class of coeffcets of dvergece of oe dstrbuto from aother, J. R. Stat. Soc. B ] A.R. Barro, L. Györf ad E.C. va der Meule, Dstrbuto estmates cosstet total varato ad two types of formato dvergece, IEEE Tras. Iform. Theory

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