H. J. Brascamp and E. H. Lieb. On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. Journal of Functional Analysis, 22:366--389, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....out that when V is a strictly convex function then (1.5) holds. Indeed, in this case a general argument based on the Bakry Emery criterium applies, see [7, 10] More directly, when there is no perturbation ( 0) Theorem 1. 1 becomes an immediate consequence of the Brascamp Lieb inequality [5]. On the other hand the extension to bounded perturbations of a strictly convex function proved to be rather challenging. We refer the reader to [4, 12, 15] and references therein to get an idea of the diculties one has to face when leaving the purely convex setting. Recently it was shown in [14] ....

....3.2. For every N 0 2 (N 0 ; 1 : 3.17) Proof. Let N; denote the canonical measure obtained in (1.2) where the potential V is replaced by its convex component . Let al..so (N; denote the corresponding Poincar e constant. Since 0 one can use the Brascamp Lieb inequality [5] to prove (N; 6 , uniformly in N and , see also [7] A standard argument (as in the proof of Lemma 2.1) on the other hand gives (N; 6 e (N; for every N 2 N and 2 R. This gives, uniformly in (N; 6 4. Ginzburg Landau processes We consider the discrete lattice ....

H. J. Brascamp, E. H. Lieb, On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diusion equation. J. Funct. Anal. 22, 366-389, 1976

....u by requiring u L 2 = 1 and u # 0, then the set D has the form, D = x : u(x) t for some t 0. Chanillo et al. show that for small enough #, all of the upper level sets of u are convex. For # = 0, this result was obtained (in a slightly stronger form) by Brascamp and Lieb (see [BL]) It turns out that the question of uniqueness is much more delicate than that of existence. Chanillo, Grieser, and Kurata demonstrate the following striking result: There are domains,# , which enjoy more symmetries than the corresponding optimal pairs. In particular, this symmetry breaking ....

H.J. Brascamp and E. Lieb, On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the di#usion equation, J. Funct. Analysis, 22(1976), 366--389.

....implies that the variance of 0 is N=2 O( p N ) 3. Upper Bound at finite fi We adapt the decimation procedure introduced in [BLL] in the case of the interface orthogonal to the vector (1; 1; 0; 0) The key step is contained in the following Theorem based on Brascamp Lieb inequality [BL] Theorem 3.1. BLL] Let : R R be a convex function such that there are positive constants A; B; C; D;E 1. 0 A 6 00 (x) 6 B 1; 8jxj E. 2. j(x) Gamma Cx 2 j 6 D 1; 8x. Then the function g defined by g(y 1 ; y k ) Gamma ln Z R exp Gamma k X i=1 (y i ....

H. Brascamp, E. Lieb, On extensions of the Brunn-Minkowski and Pr'ekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 , no. 4, 366--389 (1976).

....[16] Brooks [6] and others. The question of the extent to which the eigenvalues of the Laplace operator characterize a compact manifold has been investigated by Yau [30] Sunada [31] Brooks [7, 8, 9] Gordon, Webb and Wolpert [24] just to name a few. Numerous related results can be found in [3, 5, 11, 15, 22, 23, 27]. The ideas developed in this paper have their roots in results of the continuous setting which have been contributed by numerous people. For example, the early work of Payne, Polya and Weinberger [28] used geomtric arguments to develop quite general bounds on eigenvalue gaps. Hile and Protter # ....

H. Brascamp and E. Lieb, On extensions of the Brunn-Minkowski and Prekopa-Leindler Theorems, including inequalities for log concave functions and with an application to di#usion equation, Journal of Functional Analysis 22 (1976), 366-389.

....)B 2 is Borel. For r = 0 and r = Gamma1, is also called logarithmic concave and quasi concave, respectively ( 30] Since m r (a; b; is increasing in r if all the other variables are fixed, the sets of all r concave probability measures are increasing if r is decreasing. It is known (cf. 7] [9], 30] 31] that 2 P(IR s ) is r concave for some r 2 [ Gamma1; 1=s] if has a density f such that f (z (1 Gamma ) z) m r(s) f (z) f ( z) 6) where r(s) r(1 Gamma rs) Gamma1 holds for all 2 [0; 1] and z; z 2 IR s . We mention that e.g. the uniform distribution (on ....

H.J. Brascamp and E.H. Lieb, "On extensions of the Brunn-Minkowski and Pr'ekopaLeindler theorems, including inequalities for log concave functions and with an application to the diffusion equation," Journal of Functional Analysis 22 (1976) 366-389.

....and any fi fi 0 , we have E T L;fi (u ; u) 2 fi 1 C fi 1=2 e GammafiH ru fi fi fi fi i H 00 (0) j Gamma1 e GammafiH ru AE H (1) 2 fi 1 C fi 1=2 1 min E L;fi jruj 2 : I. 29) We compare this result to the Brascamp Lieb inequality [2, 13, 7, 11], which states that E T L;fi (u ; u) 2 fi e GammafiH ru fi fi fi fi i H 00 (x) j Gamma1 e GammafiH ru AE H (1) I.30) for strictly convex H, i.e. H 00 (x) min (x) 0, for all x 2 R j L j , where min (x) may become very small, for certain values of x. Our result ....

H.J. Brascamp and E.H. Lieb. On extensions of the Brunn-Minkowski and Pr'ekopa-Leindler theorems including inequalities for log concave functions, and with applications to the diffusion equation. J. Funct. Anal., 22:366--389, 1976.

....2 IR s , and for r 2 ( Gamma1; 0) the inequality implies that the distribution function F has the property that the extended real valued function F r is convex on IR s . Moreover, 13) and (14) imply that F is quasi concave on IR s . A useful criterion for r concavity is known from [2] [3], 10] for r = 0) and [11] It says that a measure 2 P(IR s ) is r concave for some r 2 [ Gamma1; s Gamma1 ] if has a density f such that f (y (1 Gamma ) y) m r(s) f (y) f ( y) holds for all 2 [0; 1] and y; y 2 IR s where r(s) r(1 Gamma rs) Gamma1 . For example, ....

H.J. Brascamp and E.H. Lieb, "On extensions of the Brunn-Minkowski and Pr'ekopaLeindler theorems, including inequalities for log concave functions and with an application to the diffusion equation," Journal of Functional Analysis 22 (1976) 366-389.

....(1 Gamma t)y) f(x) t f(y) 1 Gammat ; i.e log f is concave on its support. Note that the indicator functions of convex sets are log concave and that log concave functions are quasi concave. We also will need the following deep result of Pr ekopa and Leindler. Theorem( Le] and [Pr] see also [BL]) If f is log concave on IR n and 1 k n, then the function g : IR k IR , with g(x 1 ; x k ) Z IR n Gammak f(x 1 ; x k ; z 1 ; z n Gammak ) dz 7 is also log concave. Since h ffi A is log concave whenever h is log concave and A is linear, and since the ....

H. J. Brascamp and E. H. Lieb, On the extensions of the Brunn-Minkowski and Pr'ekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), 366--89.

....over many pairs (x, y) satisfying z = 1 #)x #y rather than just the pair (z, z) Though it generalises the Brunn Minkowski inequality, the Prekopa Leindler inequality is a good deal simpler to prove, once properly formulated. The argument we shall use seems to have appeared first in [Brascamp and Lieb 1976b] The crucial point is that the passage from sets to functions allows us to prove the inequality by induction on the dimension, using only the one dimensional case. We pay the small price of having to do a bit extra for this case. Proof of the Pr ekopa Leindler inequality. We start by checking ....

H. J. Brascamp and E. H. Lieb, "On extensions of the Brunn--Minkowski and Prekopa--Leindler theorems, including inequalities for log concave functions, and with an application to the di#usion equation", J. Funct. Anal. 22 (1976), 366--389.

....e ffjxj 2 = P (ffjxj 2 ) p =p . Now we consider discretization of measures similarly as (3.3) Since , s and w are convex functions, these discretized measures have log concave densities. Hence these inequalities for these discretized measures follow from Brascump Lieb inequality (see [1] Theorem 5.1) Hence taking the limit n 1 completes the proof. Combining these lemmas we see the proof of Theorem 2.1 is reduced to the bound on expectations: Let C 2:1 = supfE s ;0 T;j h exp ffjX t Gamma E s ;0 T;j [X t ]j i ; jtj T 1g C 2:2 = supfE s ;0 T;j h exp 2ffjX t ....

BRASCAMP, H.J. and LIEB, E.H. (1976). On extensions of the Brunn-- Minkowski and Pr'ekopa--Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 366--389.

....Let the charge distribution ae satisfy (C) and the potential V the condition (P 2) Then H has a unique ground state Omega 2 F , H Omega = E Omega : 1.14) Remark: P 2) means that V is strictly convex. This is a technical assumption which allows us to use the Brascamp Lieb inequality [7]. We refer to Section 4 for a discussion. Proposition 1.2. Let the charge distribution ae satisfy (C) and the potential V the conditions (P 1) P 2) Then H Gamma E and H 0 Gamma E 0 are unitarily equivalent. In particular H has the spectrum [E; 1) which is purely absolutely continuous ....

H.J. Brascamp, E.H. Lieb, On extensions of the Brunn-Minkowski and Pr'ekopa-Leindler theorems, including inequalities for log concave functions, with an application to the diffusion equation. J. Funct. Anal. 22, 366-389 (1976)

....(1 Gamma t)y) f(x) t f(y) 1 Gammat ; i.e log f is concave on its support. Note that the indicator functions of convex sets are log concave and that log concave functions are quasi concave. We also will need the following deep result of Pr ekopa and Leindler. Theorem( Le] and [Pr] see also [BL]) If f is log concave on IR n and 1 k n, then the function g : IR k IR , with g(x 1 ; x k ) Z IR n Gammak f(x 1 ; x k ; z 1 ; z n Gammak ) dz 8 G. Schechtman, Th. Schlumprecht and J. Zinn is also log concave. Since h ffi A is log concave whenever h is ....

H. J. Brascamp and E. H. Lieb, On the extensions of the Brunn-Minkowski and Pr'ekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), 366--89.

.... (1 Gamma ff)y) ffV (x) 1 Gamma ff)V (y) x; y 2 (a; b) 0 ff 1. Moreover, V is an absolutely continuous function, V 0 (x) 0. Therefore, there exist the limits lim x a V (x) V (a) lim x b Gamma V (x) V (b) and, in addition, if a or b = Sigma1, then lim x Sigma1 V (x) 1 (see [7], 8] Set V (x) 1, e GammaV (x) 0 outside the segment [a; b] For such a measure we shall calculate kd h k and k h Gamma k. It is easy to see that d 1 = GammaV 0 (x)e GammaV (x) dx e GammaV (a) ffi(a) Gamma e GammaV (b) ffi(b) whence kd 1 k = b Z a jV 0 ....

....e GammaW (x1 ; xn ) dx 2 : dx n dx 1 = Z R 1 e GammaU (x1 ) dx 1 = 1 Z 0 e GammaU (x1 ) dx 1 = 1 by the construction of the function W and the Fubini theorem. Here e GammaU (x1 ) Z R n Gamma1 e GammaW (x1 ;x2 ; xn ) dx 2 : dx n : It has been shown in [7] that the function U(x 1 ) in this case is also convex. Moreover, the function e GammaU (x1 ) does not increase on [0; 1) since all the functions e GammaW (x1 ;x 0 2 ; x 0 n ) do not increase for any fixed (x 0 2 ; x 0 n ) Therefore, we can consider the convex measure = ....

Brascamp H.J, Lieb E.H., On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log-concave functions, and with an application to the diffusion equation, J. Funct. Anal. 22 (1976), no. 4, 366--389.

....probabilities of Minkowski sums of sets. In this paper, however, we shall only consider r concave disribution functions. The concept of a log concave probabiltiy measure (the case r = 0) was intruduced and studied in [14, 15] The notion of r concavity and corresponding results were given in [2, 3]. For detailed description and proofs, see [20] By monotonicity, r concavity of a distribution function is equivalent to the inequality F (z) m r (F (x) F (y) for all z x (1 Gamma )y. Clearly, distribution functions of integer random variables are not continuous, and cannot be ....

H.J. Brascamp and E.H. Lieb, On Extensions of the Brunn--Minkowski and Pr'ekopa-- Leindler Theorems, Including Inequalities for Log-Concave Functions and with an Application to the Diffusion Equation, Journal of Functional Analysis 22 (1976) 366--389.

....from nonsmooth analysis. In order to explain this extension, we record now a metric regularity result for a situation where the data satisfy certain convexity properties. For this purpose and for later use we introduce the notion of an r concave probability measure (r 2 [ Gamma1; 1] Following [10] and [12] we define the generalized mean function m r on IR Theta IR Theta [0; 1] as follows: m r (a; b; 8 : a r (1 Gamma )b r ) 1=r if r 2 (0; 1) or r 2 ( Gamma1; 0) ab 0 0 if ab = 0; r 2 ( Gamma1; 0) a b 1 Gamma if r = 0 maxfa; bg if r = ....

....2 is Borel. For r = 0 and r = Gamma1, is also called logarithmic concave and quasi concave, respectively ( 37] Since m r (a; b; is increasing in r if all the other variables are fixed, the sets of all r concave probability measures are increasing if r is decreasing. It is known (cf. 8] [10], 37] 38] that 2 P(IR s ) is r concave for some r 2 [ Gamma1; 1=s] if has a density f such that f (z (1 Gamma ) z) m r(s) f (z) f ( z) where r(s) r(1 Gamma rs) Gamma1 (5) holds for all 2 [0; 1] and z; z 2 IR s . A density f satisfying (5) is called r(s) ....

H.J. Brascamp and E.H. Lieb, "On extensions of the Brunn-Minkowski and Pr'ekopaLeindler theorems, including inequalities for log-concave functions and with an application to the diffusion equation," Journal of Functional Analysis 22 (1976) 366-389.

....probabilities of Minkowski sums of sets. In this paper, however, we shall only consider r concave disribution functions. The concept of a log concave probabiltiy measure (the case r = 0) was intruduced and studied in [14, 15] The notion of r concavity and corresponding results were given in [2, 3]. For detailed description and proofs, see [20] By monotonicity, r concavity of a distribution function is equivalent to the inequality F (z) m r (F (x) F (y) RRR 29 98 Page 5 for all z x (1 Gamma )y. Clearly, distribution functions of integer random variables are not continuous, and ....

H.J. Brascamp and E.H. Lieb, On Extensions of the Brunn--Minkowski and Pr'ekopa-- Leindler Theorems, Including Inequalities for Log-Concave Functions and with an Application to the Diffusion Equation, Journal of Functional Analysis 22 (1976) 366--389. Page 16 RRR 29-98

....Explicit criteria for the Witten Laplacians are found from compactness of embeddings and from the Weyl calculus, which give results for closed range, strict positivity, essential self adjointness and domain characterisations. 1. Introduction and Main Results In 1976, H. J. Brascamp and E. H. Lieb [BL76] proved the following inequality for an arbitrary function f in C 1 (R n ) L 2 ( when the potential Phi is a given real, strictly convex C 2 function and when hfi = R fe Gamma Phi dx denotes the mean value of f with respect to the measure = e Gamma Phi dx: Z R n ....

H. J. Brascamp and E. H. Lieb, On extensions of the Brunn--Minkowski and Pr'ekopa--Leindler theorems including inequalities for log concave functions, and with applications to the diffusion equation, J. Funct. Analysis 22 (1976), 366--389.

....Explicit criteria for the Witten Laplacians are found from compactness of embeddings and from the Weyl calculus, which give results for closed range, strict positivity, essential self adjointness and domain characterisations. 1. Introduction and Main Results In 1976, H. J. Brascamp and E. H. Lieb [BL76] proved the following inequality for an arbitrary function f in C 1 (R n ) L 2 ( when the potential Phi is a given real, strictly convex C 2 function and when hfi = R fe Gamma Phi dx denotes the mean value of f with respect to the measure = e Gamma Phi dx: Z R n ....

H. J. Brascamp and E. H. Lieb, On extensions of the Brunn--Minkowski and Pr'ekopa--Leindler theorems including inequalities for log concave functions, and with applications to the diffusion equation, J. Funct. Analysis 22 (1976), 366--389.

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H. J. Brascamp and E. H. Lieb. On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. Journal of Functional Analysis, 22:366--389, 1976.

No context found.

Brascamp, H.J. and Lieb, E.H. (1976) On extensions of the Brunn-Minkowski and Pr'ekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal. 22, 366--389.

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H. Brascamp and E. Lieb, On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diusion equation, J. Functional Analysis 22 (1976), 366-389, MR 56#8774.

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Brascamp, H. J.; Lieb, E. H. On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diusion equation, J. Funct. Anal. 22, 366-389, (1976).

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