Rényi Divergence and L p -affine surface area for convex bodies

Transcription

1 Réyi Divergece ad L p -affie surface area for covex bodies Elisabeth M. Werer Abstract We show that the fudametal objects of the L p -Bru-Mikowski theory, amely the L p -affie surface areas for a covex body, are closely related to iformatio theory: they are expoetials of Réyi divergeces of the coe measures of a covex body ad its polar. We give geometric iterpretatios for all Réyi divergeces D, ot just for the previously treated special case of relative etropy which is the case =. Now, o symmetry assumptios are eeded ad, if at all, oly very weak regularity assumptios are required. Previously, the relative etropies appeared oly after performig secod order expasios of certai expressios. Now already first order expasios makes them appear. Thus, i the ew approach we detect faster details about the boudary of a covex body. Itroductio. There exists a fasciatig coectio betwee covex geometric aalysis ad iformatio theory. A example is the close parallel betwee geometric iequalities for covex bodies ad iequalities for probability desities. For istace, the Bru-Mikowski iequality ad the etropy power iequality follow both i a very similar way from the sharp Youg iequality see. e.g., []. Keywords: Réyi Divergece, relative etropy, L p-affie surface area. 00 Mathematics Subject Classificatio: 5A0, 53A5 Partially supported by a NSF grat, a FRG-NSF grat ad a BSF grat

2 I several recet papers, Lutwak, Yag, ad Zhag [5, 7, 9, 30] established further coectios betwee covexity ad iformatio theory. For example, they showed i [7] that the Cramer-Rao iequality correspods to a iclusio of the Legedre ellipsoid ad the polar L -projectio body. The latter is a basic otio from the L p -Bru-Mikowski theory. This L p - Bru-Mikowski theory has its origis i the 960s whe Firey itroduced his L p -additio of covex bodies. It evolved rapidly over the last years ad due to a umber of highly ifluetial works see, e.g., [5], [7] - [], [3], [4], [7] - [4], [6], [8], [3] - [34], [37], [38] - [46], [50], is ow a cetral part of moder covex geometry. I fact, this theory redirected much of the research about covex bodies from the Euclidea aspects to the study of the affie geometry of these bodies, ad some questios that had bee cosidered Euclidea i ature tured out to be affie problems. For example, the famous Busema-Petty Problem fially laid to rest i [4, 6, 48, 49], was show to be a affie problem with the itroductio of itersectio bodies by Lutwak i [4]. Two fudametal otios withi the L p -Bru-Mikowski theory are L p -affie surface areas, itroduced by Lutwak i the groud breakig paper [3] ad L p -cetroid bodies itroduced by Lutwak ad Zhag i [3]. See Sectio 3 for the defiitio of those quatities. Based o these quatities, Paouris ad Werer [35] established yet aother relatio betwee affie covex geometry ad iformatio theory. They proved that the expoetial of the relative etropy of the coe measure of a symmetric covex body ad its polar equals a limit of ormalized L p -affie surface areas. Moreover, also i [35], Paouris ad Werer gave geometric iterpretatios of the relative etropy of the coe measures of a sufficietly smooth, symmetric covex body ad its polar. I this paper we show that the very core of the L p -Bru-Mikowski theory, amely the L p -affie surface areas itself, are cocepts of iformatio theory: They are expoetials of Réyi divergeces of the coe measures of a covex body ad its polar. This idetificatio allows to traslate kow properties from oe theory to the other. Eve more is gaied. Geometric iterpretatios for all Réyi divergeces D of coe measures of a covex body ad its polar are give for all, ot just for the special case of relative etropy which correspods to the case =. We refer to Sectios ad 3 for the defiitio of D. No symmetry assumptios o K are eeded. Nor do these ew geometric iterpretatios require the strog smoothess assumptios of [35]. I the cotext of the L p -cetroid bodies, the relative etropies appeared oly after performig secod order expasios of certai expressios. The

3 remarkable fact ow is that i our approach here, already first order expasios makes them appear. Thus, these bodies detect faster details of the boudary of a covex body tha the L p -cetroid bodies. The paper is orgaized as follows. I Sectio we itroduce Réyi divergeces for covex bodies ad describe some of their properties. We also itroduce L p -affie surface areas ad mixed p-affie surface areas. The mai observatios are Theorems.4 ad.5 which show that L p - affie surface areas ad mixed p-affie surface areas are expoetials of Réyi divergeces. These idetificatios allow to traslate kow properties from oe theory to the other - this is doe i the rest of Sectio ad i Sectio 3. Also, i Sectio 3, we give geometric iterpretatios for Réyi divergeces D of coe measure of covex bodies for all, icludig ew oes for the relative etropy ot requirig the previously ecessary strog smoothess ad symmetry assumptios o the body. Further Notatio. Throughout the paper, we will assume that the cetroid of a covex body K i R is at the origi. We work i R, which is equipped with a Euclidea structure,. We deote by the correspodig Euclidea orm. B x, r is the ball cetered at x with radius r. We write B = B 0, for the Euclidea uit ball cetered at 0 ad S for the uit sphere. Volume is deoted by or, if we wat to emphasize the dimesio, by vol d A for a d-dimesioal set A. K = {y R : x, y for all x K} is the polar body of K. For a poit x K, the boudary of K, N K x is the outer uit ormal i x to K ad κ K x is the geeralized Gauss curvature i x. We write K C+, if K has C boudary K with everywhere strictly positive Gaussia curvature κ K. µ K is the usual surface area measure o K. σ is the usual surface area measure o S. Let K be a covex body i R ad let u S. The h K u is the support fuctio of directio u S, ad f K u is the curvature fuctio, i.e. the reciprocal of the Gaussia curvature κ K x at this poit x K that has u as outer ormal. Réyi divergeces for covex bodies. Let X, µ be a measure space ad let dp = pdµ ad dq = qdµ be probability measures o X that are absolutely cotiuous with respect to the 3

4 measure µ. The the Réyi divergece of order, itroduced by Réyi [36] for > 0, is defied as D P Q = log p q dµ, It is the covetio to put p q = 0, if p = q = 0, eve if < 0 ad >. The itegrals p q dµ X are also called Helliger itegrals. See e.g. [6] for those itegrals ad additioal iformatio. Usually, i the literature, 0. However, we will also cosider < 0, provided the expressios exist. We ormalize the measures as, agai usually i the literature, the measures are probability measures. Special cases. i The case = is also called the Kullback-Leibler divergece or relative etropy from P to Q see []. It is obtaied as the limit as i ad oe gets D KL P Q = D P Q = lim D P Q = p log p dµ. 3 X q The limit may ot exist but limit exists [5]. ii The case = 0 gives for q 0 with the covetio that 0 0 = that X D 0 P Q = 0, 4 as dq = qdµ is a probability measure o X. If q = 0, the D 0 P Q =. iii The case = gives D P Q = D Q P = log X p q dµ. 5 The expressio X p q dµ is also called the Bhattcharyya coefficiet or Bhattcharyya distace of p ad q. iii The cases = ad =. D P Q = log 4 sup ess px, 6 x qx

5 ad D P Q = sup ess qx x px = D Q P. 7 Note that for all,, D Q P = D P Q. 8 As, the limit o the left ad the limit o the right of 8 exist ad are equal ad equal to D Q P = X q log q p dµ. Thus 8 holds for all. We will ow cosider Réyi divergece for covex bodies K i R. Let p K x = κ K x x, N K x K, q Kx = x, N Kx. 9 K The P K = p K µ K ad Q K = q K µ K 0 are probability measures o K that are absolutely cotiuous with respect to µ K. Recall that the ormalized coe measure cm K o K is defied as follows: For every measurable set A K cm K A = } K { ta : a A, t [0, ]. The ext propositio is well kow. See e.g. [35] for a proof. It shows that the measures P K ad Q K defied i 0 are the coe measures of K ad K. N K : K S, x N K x is the Gauss map. Propositio.. Let K a covex body i R. Let P K ad Q K be the probability measures o K defied by 0. The Q K = cm K, or, equivaletly, for every measurable subset A i K Q K A = cm K A. If K is i additio i C +, the P K = N K N K cm K 5

6 or, equivaletly, for every measurable subset A i K P K A = cm K N K NK A. For =, the relative etropy of a covex body K i R was cosidered i [35], amly D P K Q K = D KL P K Q K κ K x = K x, N K x log K K K κ K x K x, N K x + dµ K x D Q K P K = D KL Q K P K x, N K x K x, N K x + = log dµ K x, K K κ K x provided the expressios exist. We ow defie the Réyi divergece of K of order for all other,,. Defiitio.. Let K be a covex body i R ad let < <,. The the Réyi divergeces of order of K are D Q K P K = κ K dµ K log K x,n K x + K K ad D P K Q K = log K κ K dµ K x,n K x + K K D Q K P K = log sup ess K x, N K x + x K K κ K x D P K Q K = log sup ess x K K κ K x K x, N K x D Q K P K = D P K Q K, D P K Q K = D Q K P K, 6 provided the expressios exist. 6

7 Remarks. i By 8 for all,, D Q K P K = D P K Q K. This idetity also holds for. Therefore, it is eough to cosider oly oe of the two, D Q K P K or D P K Q K. ii If we put N K x = u S, the x, N K x = h K u. If K is i C+, the dµ K = f K dσ. Hece, i that case, we ca express the Réyi divergeces also as D Q K P K = f K u S dσu log h K u + K K 7 D P K Q K = f K u S dσu log h K u + K K 8 Accordigly for D KL Q K P K ad D KL P K Q K. Let K,... K be covex bodies i R. Let u S. For i, defie p Ki u = ad measures o S by K i h Ki u, q K i u = f K i u h Ki u. 9 K i P Ki = p Ki σ ad Q Ki = q Ki σ. 0 The we defie the Réyi divergeces of order for covex bodies K,... K by Defiitio.3. Let K,... K be covex bodies i R. The for < <, f log Ki h K i dσ D Q K Q K P K P K = 7 S i= K i Ki

8 D P K P K Q K Q K = log provided the expressios exist. For = the defiitios were give i [35]: D Q K Q K P K P K = f Ki h Ki log S K i i= f K h i K i S i= K i Ki i= K i f Ki h + K i K i dσ dσ D P K P K Q K Q K = h K i S Ki log i= provided the expressios exist. i= K i K i f Ki h + K i dσ, Remark. For < <,, D P K P K Q K Q K = D Q K Q K P K P K, ad, agai, for, the limits o both sides exist ad coicide. Therefore it is eough to cosider either D P K P K Q K Q K or D Q K Q K P K P K. We first preset some examples ad look at special cases below. I particular, D ± Q K Q K P K P K will be cosidered below. Examples. i If K = ρb, the D Q K P K = D P K Q K = 0 for all. ii If K is a polytope, the κ K = 0 a.e. o K. Thus, for =, κ D Q K P K =. For < <, K dµ K = 0 ad K x,n K x + for >, κ K dµ K K =. Hece D x,n x + Q K P K = for all < <, ad K a polytope. 8

9 Similarly, D P K Q K = 0 with the covetio that 0 = 0. D P K Q K =, for < < ad < < 0 ad K a polytope ad D P K Q K =, for 0 < < ad K a polytope. This also shows that D eed ot be cotiuous at =. For = 0 ad = ±, see below. iii For < r <, let K = Br = {x R : i= x i r } be the uit ball of lr. We will compute D Q K P K ad D P K Q K for all < <,. The case = was cosidered i [35]. The cases = 0 ad = ± are treated below. If < r < ad r, the D P B r Q B r =. If < r < ad r r, the D Q B r P B r =. If < r < ad r, the D P B r Q B r =. If < r < ad r r, the D Q B r P B r =. I all other cases we have ad D P B r Q B r = [ log Γ r Γ r Γ r Γ r Γ r + r ] Γ r + r D Q B r P B r = [ log Γ r Γ r Γ r Γ r Γ r + r ] Γ r + r. Now we itroduce L p -affie surface areas for a covex body K i R. L p -affie surface area, a extesio of affie surface area, was itroduced by Lutwak i the groud breakig paper [3] for p > ad for geeral p by Schütt ad Werer [4]. For real p, we defie the L p -affie surface area as p K of K as i [3] p > ad [4] p <, p by as p K = K κ K x p +p x, N K x p +p dµ K x ad as ± K = K κ K x x, N K x dµ Kx, 3 9

10 provided the above itegrals exist. I particular, for p = 0 as 0 K = x, N K x dµ K x = K. K The case p = is the classical affie surface area which goes back to Blaschke. It is idepedet of the positio of K i space. as K = κ K x + dµk x. K Origially a basic affie ivariat from the field of affie differetial geometry, it has recetly attracted icreased attetio too e.g. [9, 3, 3, 39, 44]. If K is i C+, the dµ K = f K dσ ad the the L p -affie surface areas, for all p, ca be writte as f K u +p as p K = dσu. 4 S h K u p +p I particular, as ± K = S dσu h K u = K. Recall that f K u is the curvature fuctio of K at u, i.e., the reciprocal of the Gauss curvature κ K x at this poit x K, the boudary of K, that has u as its outer ormal. The mixed p-affie surface area, as p K,, K, of covex bodies K i C + was itroduced - for p i [] ad exteded to all p i [47] - as as p K,, K = S [ ] h K u p f K u h p +p K f K u dσu. 5 The we observe the followig remarkable fact which coects L p -Bru Mikowki theory ad iformatio theory: L p -affie surface areas of a covex body are Helliger itegrals - or expoetials of Réyi divergeces - of the coe measures of K ad K. For =, such a coectio was already observed i [35], amely +p K K e D KLP K Q K asp K = lim. 6 p K Now we have more geerally 0

11 Theorem.4. Let K be a covex body i R. Let < <.. The D P K Q K = as log K K K. D Q K P K = as log K K K. Equivaletly, for all p, p, as p K K +p K p = Exp +p + p D p P K Q +p K = Exp p + p D Q K P +p K I particular, as K K + K + = Exp + D + P K Q K = Exp + D Q K P + K. Remarks. i Theorem.4 ca also be writte as +p asp K K = K K e D p +p P K Q K. If we ow let p, we recover 6. Also from Theorem.4 +p asp K p K If we let p 0, the we get = K K e D +p Q K P K. lim p 0 +p asp K p K = K K We will commet o these expressios i Sectio 3. e D KLQ K P K. 7 ii If < 0, the p = <. Thus, for this rage of, we get the L p -affie surface area i the rage smaller tha. If

12 0 <, the < p =. Thus, for this rage of, we get the L p -affie surface area i the rage greater tha. I particular, for 0, we get the L p -affie surface area for 0 p. If <, the < p =. Thus, for this rage of, we get the L p -affie surface area i the rage greater. If <, the p = <. Thus, for this rage of, we get the L p-affie surface area i the rage smaller tha. Theorem.5. Let K,... K be covex bodies i C+. The, for all D P K P K Q K Q K = as log K,..., K i= K i Ki ad D Q K Q K P K P K = log Remark. as K,..., K i= K i Ki The expressios i Theorem.5 ca also be writte as as K,..., K Ki i= K i = e D P K P K Q K Q K K i= i. ad as K,..., K i= K i = i= K i e D Q K Q K P K P K. K i If we ow let i the first expressio respectively, puttig p =, p, we get Ki e D P K P K Q K Q K Ki i= = lim = lim p as K,..., K i= K i as p K,..., K i= K i +p. 8.

13 If we let i the secod expressio, respectively, puttig p =, p 0, we get K i e D Q K Q K P K P K i= K i as = lim i= K i = lim p 0 K,..., K as p K,..., K i= K i +p p. 9 We will commet o these quatities i Sectio 3. Special Cases. i If =, the D Q K P K = D P K Q K = log as K K K, ad as K K K is the Bhattcharyya coefficiet of p K ad q K. D Q K Q K P K P K = = D P K P K Q K Q K as K,..., K = log i= K i K i ii If = 0, the D 0 P K Q K = 0. Likewise, as K D 0 Q K P K = log K 30 which, if K is sufficietly smooth, is equal to as K K log K = log K κ K xdµx x,n K x = log = 0 3

14 ad equal to if K is a polytope. as 0 K,..., K D 0 P K P K Q K Q K = log i= K i ad as K,..., K D 0 Q K Q K P K P K = log i= K i Ṽ K,..., K = log i= K i, where Ṽ K,..., K is the dual mixed volume itroduced by Lutwak i []. iii If, the p = from the right. Therefore, by defiitio, D Q K P K = log sup x ess q Kx p K x = log sup x ess x,n Kx + K κ K x K. O the other had lim as K K K = K K = K K lim x, N K x + κ K x x, N K x + κ K x, L L which is thus cosistet with the defiitio of D Q K P K. Similarly, oe shows that, if, the p = from the left. Hece, by defiitio, D P K Q K = log sup x ess q Kx p K x = log sup x ess, which is cosistet with lim as K K K. Thus, also it would make most sese to defie ad lim as pk = sup p + x K lim as pk = sup ess p x K which would imply that lim p as p K does ot exist. 4 κ K x K x,n K x + K ess x, N Kx +. 3 κ K x κ K x, 3 x, N K x +

15 If, the p = from the left ad by 7, D Q K P K = D P K Q K. O the other had, as lim log K K K = log sup x = log sup x = D P K Q K, κ K x K x,n K x + K κ K x K x, N K x + K hece this is also cosistet with the defiitios. Similar cosideratios hold for D P K Q K ad D P K P K Q K Q K ad D Q K Q K P K P K. Havig idetified L p -affie surface areas as Réyi divergeces, we ca ow traslate kow results from oe theory to the other. Affie ivariace of L p -affie surface areas traslates ito affie ivariace of Réyi divergeces: For all p, as p T K = det T p +p as p K see [4]. Theorem.4 the implies that for all liear maps T with det T 0, for all < <,, ad D P T K Q T K = D P K Q K D Q T K P T K = D Q K P K. The case = was treated i [35]. As as p T K,..., T K = det T p +p as p K,..., K see [47], it follows from Theorem.5 that for all liear maps T with det T 0, for all < <,, ad D P T K P T K Q T K Q T K = D P K P K Q K Q K D Q T K Q T K P T K P T K = D Q K Q K P K P K. The case = is i [35]. 5

16 Moreover, all iequalities ad results metioed i e.g. [46] about L p - affie surface area ad i e.g. [47] about mixed L p -affie surface area ca be traslated ito the correspodig iequalities ad results about Réyi divergeces. Coversely, results about Réyi divergeces from e.g. [3] have cosequeces for L p -affie surface areas. We metio oly a few. Propositio.6. Let K be a covex body i C +. i The for all, ad D Q K P K = D Q K P K D P K Q K = D P K Q K The equalities hold trivially if = 0 or =. ii Let K i, i, be covex bodies i C +. The for all 0 as K,...,, K = = S i= m i= m [f Ki h S i.e. we ca iterchage itegratio ad product. [f Ki h K i ] ] K i iii Let K ad L be covex bodies i C+. Let 0 p. Let 0 λ. The S [ λ f Kh K K + λ f Lh L L as p K K +p K ] +p [ p +p λ h K K + λ h L L λ as p L L +p L dσ dσ, ] p +p dσ p +p λ with equality iff K = L. Equality holds trivially if p = 0 or p = or λ = 0 or λ =. Proof. i For < <, i follows from the duality formula as p K = K, or, formulated i a more symmetric way, usig the parameter as p = p +p as K = as K. 6

17 This idetity was proved for p > 0 i [] ad - with a differet proof - for all other p i [46]. Let ow =. The, o the oe had lim D Q K P K = D Q K P K = log sup O the other had, by 6, ess q K x x K p K x. 33 D Q K P K = D P K Q K = log sup ess p Kx x K q K x equals 34, as see [] for x K, y K such that x, y =, Similarly, for =. y, N K y x, N K x = κ K yκ K x +. ii follows from Theorem.5 ad the fact that [3] D Q K Q K P K P K = D Q Ki P Ki, i= respectively the correspodig equatio for D P K P K Q K Q K. iii For 0, D Q K P K, respectively D P K Q K, are joitly covex [3]. We put p = respectively p = ad use the joit covexity together with Theorem.4. If p 0, ad λ 0,, the equality implies that K = L as the logarithm is strictly cocave. 3 Geometric iterpretatio of Réyi Divergece I this sectio we preset geometric iterpretatios of Réyi divergeces D of covex bodies, for all. Geometric iterpretatios for the case =, the relative etropy, were give first i [35] i terms of L p -cetroid bodies. Recall that for a covex body K i R of volume ad p, the L p -cetroid body Z p K is this covex body that has support fuctio h ZpKθ = K 7 x, θ p dx /p.

18 Now that we observed that Réyi divergeces are logarithms of L p -affie surface areas, we ca use their geometric characterizatios to obtai the oes for Réyi divergeces. We will mostly cocetrate o the geometric characterizatio of L p -affie surface areas via the surface bodies [4] ad illumiatio surface bodies [47], though there are may more available see e.g. [33, 40, 45, 46] Eve more is gaied. Firstly, we eed ot assume that the body is symmetric as i [35] or that it has C+ boudary as it was eeded i [35], to obtai the desired geometric iterpretatio for the D for all. Weaker regularity assumptios o the boudary suffice. Secodly, i the cotext of the L p -cetroid bodies, the relative etropies appeared oly after performig a secod order expasio of certai expressios. Now, usig the surface bodies or illumiatio surface bodies, already a first order expasio makes them appear. Thus, these bodies detect faster details of the boudary of a covex body tha the L p -cetroid bodies. Let K be a covex body i R. Let f : K R be a oegative, itegrable, fuctio. Let s 0. The surface body K f,s, itroduced i [4], is the itersectio of all closed half-spaces H + whose defiig hyperplaes H cut off a set of fµ K -measure less tha or equal to s from K. More precisely, K f,s = H +. R K H fdµ K s The illumiatio surface body K f,s [47] is defied as { } K f,s = x : µ f K [x, K]\K s, where for sets A ad B respectively poits x ad y i R, [A, B] = {λa+ λb : a A, b B, 0 λ } respectively [x, y] = λx + λy : 0 λ } is the covex hull of A ad B respectively x ad y. For x K ad s > 0 ad f ad K f,s as above, we put x s = [0, x] K f,s. The miimal fuctio M f : K R M f x = if 0<s K H x s,n Kf,s x s f dµ K vol K H x s, N Kf,s x s 35 8

19 was itroduced i [4]. Hx, ξ is the hyperplae through x ad orthogoal to ξ. H x, ξ is the closed halfspace cotaiig the poit x + ξ, H + x, ξ the other halfspace. For x K, we defie rx as the maximum of all real umbers ρ so that B x ρn Kx, ρ K. The we formulate a itegrability coditio for the miimal fuctio dµ K x <. 36 K M f x rx The followig theorem was proved i [4]. Theorem 3.. [4] Let K be a covex body i R. Suppose that f : K R is a itegrable, almost everywhere strictly positive fuctio that satisfies the itegrability coditio 36. The c = B. K K f,s c lim = s 0 s K κ f dµ K. Theorem 3. was used i [4] to give geometric iterpretatios of L p - affie surface area. Now we use this theorem to give geometric iterpretatios of Réyi divergece of order for all for coe measures of covex bodies. First we treat the case. Corollary 3.. Let K be a covex body i R. For p, p, let f p : K R be defied as f p x = x, N Kx p +p κ K x p p +p If f p is almost everywhere strictly positive ad satisfies the itegrability coditio 36, the c K +p K p +p K K fp,s lim = Exp s 0 s. p + p D Q K P +p K, 9

20 ad, provided p ±, c K K fp,s K +p K p lim = Exp +p s 0 s + p D p P K Q +p K. If K is i C +, the last equatio also holds for p = ±. Proof. The proof of the corollary follows immediately from Theorems 3. ad.4. The ext corollary treats the case =. There, we eed to make additioal regularity assumptios o the boudary of K. Those are weaker though tha C+. Corollary 3.3. Let K be a covex body i R. Assume that K is such that there are 0 < r R < so that for all x K B x rn K x, r K B x RN K x, R. 37 Let f P Q : K R ad f QP : K R be defied by f P Q x = K x, N K x κ K x f QP x = log K κ K x log x, N K x R K κ K x r K x, N K x + R K x, N K x + r K κ K x, The f P Q ad f QP are almost everywhere strictly positive, satisfy the itegrability coditio 36 ad. K K fp c lim Q,s s 0 s = D KL P K Q K + log R as± K r K. If K is i C+, the this equals D KL NK N K cm K cm K + log R r. K K fqp,s c lim s 0 s = D KL Q K P K + log R r If K is i C+, the this is equal to D KL NK N K cm K cm K + log R r. 0

21 Proof. Note that r = R iff K is a Euclidea ball with radius r. The the right had sides of the idetities i the corollary are equal to 0 ad f P Q ad f QP are idetically equal to. Therefore, for all s 0, K fp Q,s = K ad K fqp,s = K ad hece for all s 0, K K fp Q,s = 0 ad K K fqp,s = 0. Therefore, the corollary holds trivially i this case. Assume ow that r < R. The ad we get for all x K that Also, for all x K, R K κ K x r K x, N K x + K r f P Q x log R r K r log R r R 4, r > 0. satisfies the itegrability coditio 36. follows immediately from Theorem 3.. If K is i C+, Similarly for f QP. If K is i C+, coditio 37, holds. We ca take r = if x K mi r ix ad R = sup i M fp Q x ad therefore f P Q The proof of the corollary the x K max r ix, 38 i where for x K, r i x, i are the pricipal radii of curvature. For covex bodies K ad K i, i =,,, defie fn K u = f Ku [fp K, u f p K, u] +p, where f p K, u = h K u p f K u. Corollary 3.4. Let K ad K i, i =,,, be covex bodies i C +. The c +p i= K i Ki P Exp + p D p K K f,s lim = s 0 s P K +p P K Q K Q K,

22 ad c +p i= K i Ki P Exp p + p D K K f,s lim = s 0 s Q K +p Q K P K P K. Proof. Agai, the proof follows immediately from Theorems 3. ad.5. Remark. It was show i [47] that for a covex body K i R with C +-boudary lim c K f,s K = s 0 s K κ K x fx dµ K x, 39 where c = B ad f : K R is a itegrable fuctio such that f c µ K -almost everywhere. c > 0 is a costat. Usig 39, similar geometric iterpretatios of Réyi divergece ca be obtaied via the illumiatio surface body istead of the surface body. We ca use the same fuctios as i Corollary 3., Corollary 3.3 ad Corollary 3.4. We will also have to assume that K is i C+. I [35], the followig ew affie ivariats Ω K were itroduced ad its relatio to the relative etropies were established: Let K, K,..., K be covex bodies i R, all with cetroid at the origi. The ad ad Ω K = lim p asp K +p K. asp K,..., K +p Ω K,...K = lim. p as K,..., K It was proved i [35] that for a covex body K i R that is C+ K D KL P K Q K = log K Ω K K D KL Q K P K = log 40 K Ω K. 4

23 Note that equatio 40 also followed from 6. Similar results hold for Ω K,...K. We ow cocetrate o Ω K. As show i [35], these ivariats ca also be obtaied as Ω K = lim p 0 asp K +p p K +p aspk p ad thus, deotig by A K = lim p 0 K, Ω K = A K. This implies e.g. that lim as pk p 0 K as p K K p =. Geometric iterpretatios i terms of L p -cetroid bodies were give i [35] for the ew affie ivariats Ω K. These iterpretatios are i the spirit of Corollaries 3., 3.3 ad 3.4: As p, appropriately chose volume differeces of K ad its L p -cetroid bodies make the quatity Ω K appear. Agai, however, with the L p -cetroid bodies, oly symmetric covex bodies i C + could be hadled ad it was eeded to go to a secod order expasio for the volume differeces. Now, it follows from Corollary 3.3 that there exist such iterpretatios for Ω K also for o-symmetric covex bodies ad uder weaker smoothess assumptios tha C +. Moreover, agai already a first order expasio gives such geometric iterpretatios if oe uses the surface bodies or the illumiatio surface bodies istead of the L p -cetroid bodies. Corollary 3.5. Let K be a covex body i R such that 0 is the ceter of gravity of K ad such that K satisfies 37 of Corollary 3.3. Let f P Q : K R ad f QP : K R be as i Corollary 3.3. The ad K K fp c lim Q,s log s 0 s K K fqp,s c lim log s 0 s R r as± K K R K = log r 3 K = log K Ω K = log K Ω K = log K K A K K K A K.

24 Proof. The proof of the corollary follows immediately from Corollary 3.3, 40, 4 ad the defiitio of A K. 4

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28 [43] A. Stacu, O the umber of solutios to the discrete two-dimesioal L 0 -Mikowski problem. Adv. Math , [44] E. Werer, Illumiatio bodies ad affie surface area, Studia Math [45] E. Werer, O L p -affie surface areas, Idiaa Uiv. Math. J. 56, No , [46] E. Werer ad D. Ye, New L p affie isoperimetric iequalities, Adv. Math , o. 3, [47] E. Werer ad D. Ye, Iequalities for mixed p-affie surface area, Math. A , [48] G. Zhag, Itersectio bodies ad Busema-Petty iequalities i R 4, A. of Math , [49] G. Zhag, A positive aswer to the Busema-Petty problem i four dimesios, A. of Math , [50] G. Zhag, New Affie Isoperimetric Iequalities, ICCM 007, Vol. II, Elisabeth Werer Departmet of Mathematics Uiversité de Lille Case Wester Reserve Uiversity UFR de Mathématique Clevelad, Ohio 4406, U. S. A Villeeuve d Ascq, Frace elisabeth.werer@case.edu 8