MOMENT-ENTROPY INEQUALITIES

Erwin Lutwak, Deane Yang and Gaoyong Zhang Department of Mathematics Polytechnic University Brooklyn, NY 11201 Abstract. It is shown that the product of the R´ enyi entropies of two independent random vectors provides a sharp lower bound for the expected value of the moments of the inner product of the random vectors. This new inequality contains important geometry (such as extensions of one of the fundamental affine isoperimetric inequalities, the Blaschke-Santal´ o inequality).

Introduction Vitale (1996a, 1996b, 2001) presents evidence of a surprising connection between probability and analytic convex geometry. In this paper we contribute additional evidence of this unexpected link between these subjects. It will be shown that the product of the R´enyi entropies of two independent random vectors provides a sharp lower bound for the expected value of the moments of the inner product of the random vectors. Our new inequality encodes important geometry. For example, a non-technical version of the Blaschke-Santal´o inequality for compact sets is but one special case. This paper deals with random vectors in Euclidean n-space, Rn . For vectors x, y ∈ Rn let x·y denote their inner product. If φ ∈ GL(n), then we use φ−t to denote the inverse of the transpose of φ. If X is a random vector in Rn with density f , then for λ > 0, the λ-R´enyi entropy of X is defined (see e.g., Cover and Thomas (1992) and Gardner (2002)) by ½ Entλ (X) =

1 1−λ



R

log

Rn

R Rn

f (x)λ dx

λ 6= 1,

f (x) log f (x) dx λ = 1.

1991 Mathematics Subject Classification. 60D05, 94A17, 52A40. Key words and phrases. Blaschke-Santal´ o inequality, R´ enyi entropy, dual mixed volumes. ∗ Research supported, in part, by NSF Grant DMS–0104363 Typeset by AMS-TEX

1

2

MOMENT-ENTROPY INEQUALITIES

The λ-R´enyi entropy power of X is defined as Nλ (X) = eEntλ (X) ,

with

N∞ (X) = lim Nλ (X). λ→∞

A random vector in Rn , with density function f , has finite pth moment provided that Z |x|p f (x)dx < ∞, Rn

where |x| denotes the ordinary Euclidean norm of x ∈ Rn . For φ ∈ GL(n), and p, λ > 0, define densities φp,λ : Rn → R by  1 p λ−1 b (1 + |φx| )   p b e−|φx| φp,λ (x) =  1  b (1 − |φx|p )+λ−1

λ < 1, λ = 1, λ > 1,

where (z)+ = max{0, z}, and in each case b = bp,λ is chosen so that φp,λ is a density. We shall prove an extended version of: n Theorem. Suppose real p ≥ 1 and real λ > n+p are fixed. If X and Y are independent n th random vectors in R that have finite p moment, then p

E(|X · Y |p ) ≥ c1 [Nλ (X)Nλ (Y )] n , where the best possible c1 is given by ³ p + 1 ´ ³ n + p ´−1 ³ n ´1+ 2p n 2 c1 = Γ Γ + 1 c0 , 1 Γ 2 2 2 nπ p+ 2  ³ ´ 1 ³ ´ np ¡ ¢ n(1−λ) λ−1 pλ n 1 n n   1 − − 1 B , −  p pλ n(1−λ) p 1−λ p    ³ ´n ¡ ¢ −n/p pe p c0 = Γ np + 1 n     ³ ´ 1 ³ ´ np ¡  ¢  n n(λ−1) λ−1 pλ n λ 1 + + 1 B , p pλ n(λ−1) p λ−1 2

λ<1 λ=1 λ > 1.

Equality occurs if and only if X has density a.e. φp,λ and Y has density a.e. (aφ−t )p,λ , with a > 0. If K ⊂ Rn is a compact set with volume (i.e., Lebesgue measure) V (K) and X has density 1K /V (K), then trivially N∞ (X) = V (K). Now if K, L ⊂ Rn are compact and we let X, Y have densities 1K /V (K), 1L /V (L) then letting λ → ∞ and p → ∞ in the theorem gives the following inequality.

MOMENT-ENTROPY INEQUALITIES

3

Corollary. If K, L ⊂ Rn are compact, then ωn2 max |x · y|n ≥ V (K)V (L). x∈K,y∈L

Here ωn = π n/2 /Γ(1 + n/2) denotes the volume of the unit ball in Rn . If K is an origin-symmetric convex body and L is the polar of K, then the above inequality is the classical Blaschke-Santal´ o inequality, with sharp constant. See e.g., the books of Gardner (1995), Leichtweiß (1998), Schneider (1993), and Thompson (1996) (and also the article of Bourgain and Milman (1987)) for references regarding the BlaschkeSantal´ o inequality. 0. Dual mixed volumes of random vectors Each non-negative ρ ∈ Lq (S n−1 ) defines a star-shaped set K = Kρ ⊂ Rn by K = { ru : 0 ≤ r ≤ ρ(u) with u ∈ S n−1 }. The set K is called the Lq -star generated by ρ and the function ρ is called the radial function of K (and is often written as ρK to indicate its relationship to K). We will not distinguish between Lq -stars whose radial functions are a.e. equal. It is convenient to extend the definition of the radial function from S n−1 to Rn \ {0} by making it homogeneous of degree −1. Thus, for an Lq -star K ⊂ Rn , the radial function ρK : Rn \ {0} → [0, ∞) can be defined by ρK (x) = max{ r ≥ 0 : rx ∈ K }. From this it follows immediately that if φ ∈ GL(n) then the radial function of the star φK = {φx : x ∈ K} is given by ρφK (x) = ρK (φ−1 x) for all x 6= 0. From this, and the homogeneity (of degree −1) of the radial function, it follows immediately that E is an origin-centered ellipsoid of positive volume if and only if there exists a φ ∈ GL(n) such that 1/ρE (x) = |φx|, for all x 6= 0. Elements of the dual Brunn-Minkowski theory of Lq -stars were studied by Klain (1996, 1997). Other extensions for the dual Brunn-Minkowski theory have been considered by Gardner and Volˇciˇc (1994) and Gardner, Vedel Jensen, and Volˇciˇc (2003). In particular, the latter paper defines dual mixed volumes of bounded Borel sets in terms of moments. An Lq -star whose radial function is both positive and continuous is called a star body. A star body that is convex is called a convex body. Note that throughout, convex bodies are assumed to contain the origin in their interiors. If K is a convex body in Rn , then the polar, K ∗ , of K is defined by K ∗ = {x ∈ Rn : x·y ≤ 1 for all y ∈ K}. It follows immediately from this definition that if φ ∈ GL(n), then (φK)∗ = φ−t K ∗ . From this we see that ρ1 , ρ2 are radial functions of polar reciprocal origin-centered ellipsoids if and only if there exists a φ ∈ GL(n) such that 1/ρ1 (x) = |φx|, for all x 6= 0.

and

1/ρ2 (x) = |φ−t x|,

(0.1)

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MOMENT-ENTROPY INEQUALITIES

Suppose p > 0. Define the dual mixed volume V˜−p (K, L) of an Ln+p -star K and a star body L by: 1 V˜−p (K, L) = n

Z n+p

S n−1

ρK (u)

−p

ρL (u)

n+p du = n

Z K

ρL (x)−p dx,

(0.2)

where in the left integral the integration is with respect to Lebesgue measure on S n−1 and in the right integral the integration is with respect to Lebesgue measure on Rn . Obviously, for each star body L, V˜−p (L, L) = V (L).

(0.3)

From (0.2) and the H¨older inequality we see that if K is an Ln+p -star and L is a star body, then we have the dual mixed volume inequality: V˜−p (K, L)n ≥ V (K)n+p V (L)−p ,

(0.4)

with equality if and only if there exists a c > 0 such that, a.e., ρK = cρL . Thus for the volume of each Ln+p -star, K, we have V (K)

n+p n

= Inf{V˜−p (K, Q) : Q is a star body with V (Q) = 1}.

(0.5)

If X is a random vector in Rn , that has finite pth moment and L is star body in Rn , then define the dual mixed volume V˜−p (X, L) by n+p V˜−p (X, L) = n

Z Rn

ρL (x)−p f (x) dx,

(0.6)

where f is the density function X. It will be on occasion convenient to write V˜−p (f, L) rather than V˜−p (X, L). For p, λ > 0, define pλ : R+ → R+ by  1 p λ−1   (1 +p s ) e−s pλ (s) =  1  (1 − sp )+λ−1

λ < 1, λ = 1, λ > 1.

We shall use: Lemma 0.1. Suppose K is a star body in Rn . For real a, p, λ > 0, with λ > n/(n+p), Z Rn

ρ−p (x)pλ (a/ρK (x)) dx = a−(n+p) α1 V (K), K

MOMENT-ENTROPY INEQUALITIES

where Z



α1 = n

sn+p−1 pλ (s) ds =

0

   

n+p n λ n p B( p , 1−λ − p ) n+p n p Γ( p ) n+p n λ p B( p , λ−1 )

  

5

λ < 1, λ = 1,

(0.1.1)

λ > 1.

Proof. Rewrite the integral over Rn as an integral over S n−1 × (0, ∞) Z Z Z ∞ −p ρ−p (ru)pλ (a/ρK (ru))rn−1 drdu, ρK (x)pλ (a/ρK (x)) dx = K Rn

S n−1

0

and observe that the inner integral is easily evaluated. Specifically, for fixed u ∈ S n−1 , make the change of variable s = [a/ρK (u)]r = a/ρK (ru) and observe Z ∞ Z ∞ −p n−1 −(n+p) n ρK (ru)pλ (a/ρK (ru))r dr = a ρK (u) sn+p−1 pλ (s) ds 0

=

0 −(n+p) 1 n α1 a n ρK (u).

¤

Thus, if K is a star body and b is chosen so that bpλ (a/ρK ) is a probability density and a > 0, then V˜−p (bpλ (a/ρK ), K) = ba−(n+p) α1 V (K). We shall need: Lemma 0.2. Suppose K is a star body in Rn . For a, p, λ > 0, with λ > n/(n + p), Z pλ (a/ρK (x)) dx = a−n α2 V (K), Rn

where Z



α2 = n

sn−1 pλ (s) ds =

0

    

n 1 n n p B( p , 1−λ − p ) n n p Γ( p ) n n λ p B( p , λ−1 )

λ < 1, λ = 1,

(0.2.1)

λ > 1.

The proof is similar to that of Lemma 0.1. We shall also use: Lemma 0.3. Suppose K is a star body in Rn . For a, p, λ > 0, with λ > n/(n + p), Z pλ (a/ρK (x))λ dx = a−n α3 V (K), Rn

where Z



α3 = n 0

sn−1 pλ (s)λ ds =

    

The proof is similar to that of Lemma 0.1.

n n λ p B( p , 1−λ n n p Γ( p ) n n λ p B( p , λ−1

− np )

λ < 1, λ = 1,

+ 1)

λ > 1.

(0.3.1)

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MOMENT-ENTROPY INEQUALITIES

´nyi entropy 1. Constrained maximum Re The following lemma presents the solution to the problem of maximizing the λ-R´enyi entropy when the value of the dual mixed volume of a random vector is fixed. Lemma 1.1. Suppose K is a star body in Rn and real p, λ, c > 0, with λ > Consider the problem of finding

n n+p .

max Entλ (X), subject to the constraint that X be a random vector in Rn , with finite pth moment, such that V˜−p (X, K) = c. Then, the unique maximum is achieved by the random vector whose density function is a.e. bpλ (a/ρK ), where b > 0 is chosen so that bpλ (a/ρK ) is a probability density and a > 0 is chosen so that V˜−p (bpλ (a/ρK ), K) = c. Proof. Suppose f is a probability density on Rn such that V˜−p (f, K) = c. For the sake of notational simplicity let M denote Z n ˜ V−p (f, K) = ρ−p f dx, M= K n+p Rn and let gλ,p = bpλ (a/ρK ), where b > 0 is chosen so that bpλ (a/ρK ) is a probability density and a > 0 is chosen so that Z Z Z −p −p M =b ρK pλ (a/ρK ) dx = ρK gλ,p dx = ρ−p f dx. (1.1.1) K Rn

Rn

Rn

(I) Case λ = 1. p

p

From the fact that g1,p = be−a /ρK , and the fact that f and g1,p are probability densities, together with the last identity in (1.1.1), and the definition of Ent1 , we have Z Z Z f log(g1,p /f ) dx = − f log f dx + f log g1,p dx Rn Rn Rn Z Z =− f log f dx + (log b − ap ρ−p )f dx K n n R R Z Z =− f log f dx + (log b − ap ρ−p )g1,p dx K Rn

Rn

= Ent1 (f ) − Ent1 (g1,p ).

MOMENT-ENTROPY INEQUALITIES

From the strict concavity of the log function, we see that Z Z Z f log(g1,p /f ) dx ≤ log g1,p dx ≤ log Rn

Rn

Rn

7

g1,p dx = 0,

with equality if and only if a.e. f = g1,p . Thus Ent1 (f ) ≤ Ent1 (g1,p ), with equality if and only if a.e. f = g1,p . (II) Case λ 6= 1. From the fact that gλ,p is a density function, Lemmas 0.2, 0.3, and 0.1, we have Z gλ,p dx = ba−n α2 V (K) 1= n ZR λ gλ,p dx = bλ a−n α3 V (K) n R Z ρ−p M= gλ,p dx = ba−(n+p) α1 V (K). K Rn

Thus

Z 1−λ

b

Rn

λ dx = gλ,p

α3 , α2

and

ap M =

α1 . α2

(1.1.2)

We now divide the case λ 6= 1 into two subcases, sub-case λ < 1 and sub-case λ > 1. (II 1 ) Sub-case λ < 1. The H¨older inequality and the fact that f and gλ,p are probability densities, shows that µZ ¶1−λ µZ ¶λ Z 1−λ λ gλ,p f dx ≤ gλ,p dx f dx = 1, (1.1.3) Rn

Rn

Rn

with equality if and only if a.e. f = gλ,p . 1−λ From the definition of gλ,p and the definition of pλ , for λ < 1, we see that gλ,p = 1−λ p −p 1−λ b − a ρK gλ,p . From this and the H¨older inequality, again, we have: Z Z Z 1−λ λ 1−λ λ 1−λ λ gλ,p f dx = b f dx − ap ρ−p gλ,p f K Rn

Rn

Rn

1−λ

λ

≥b

f dx − a

p

Rn

−p

−p

Rn

Z 1−λ

¶1−λ µZ

µZ

Z

ρK gλ,p dx

Rn

ρK

¶λ f dx

f λ dx − ap M.

=b

Rn

This together with (1.1.3) shows that Z 1−λ f λ dx ≤ ap M + 1, b Rn

(1.1.4)

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MOMENT-ENTROPY INEQUALITIES

with equality if and only if a.e. f = gλ,p . In this sub-case (1.1.2), together with (0.1.1), (0.2.1) and (0.3.1) give: Z λ α3 1−λ 1−λ λ b gλ,p dx = = λ n α2 Rn 1−λ − p α1 a M= = α2

n p

p

It follows that

λ 1−λ



n p

.

Z p

1−λ

a M +1=b

Rn

This and (1.1.4) gives the desired result that Z Z λ f dx ≤ Rn

Rn

λ gλ,p dx.

λ gλ,p dx,

with equality if and only if a.e. f = gλ,p . (II 2 ) Sub-case λ > 1. From the the H¨older inequality we have µZ ¶1−1/λ µZ Z λ−1 λ gλ,p f dx ≤ gλ,p dx Rn

Rn

¶1/λ λ

f dx

,

(1.1.5)

Rn

with equality if and only if a.e. f = gλ,p . From the definition of gλ,p and the definition of pλ , for λ > 1, we see that in this λ−1 case gλ,p ≥ bλ−1 (1 − ap ρ−p ). This and the fact that f is a probability density gives: K µ ¶ Z Z λ−1 λ−1 p −p gλ,p f dx ≥ b 1−a ρK f dx = bλ−1 (1 − ap M ). (1.1.6) Rn

Rn

In this sub-case (1.1.2), together with (0.1.1), (0.2.1) and (0.3.1) give: Z λ α3 λ−1 1−λ λ b gλ,p dx = = λ n α2 Rn λ−1 + p ap M =

α1 = α2

These identities yield

n p λ λ−1

+

Z λ−1

b

p

(1 − a M ) = Rn

and thus from (1.1.6) we have: Z Rn

λ gλ,p dx,

Z λ−1 gλ,p f

dx ≥ Rn

This and (1.1.5) gives the desired result: Z Z λ f dx ≥ Rn

with equality if and only if a.e. f = gλ,p .

g λ dx, Rn

¤

λ gλ,p dx.

n p

.

MOMENT-ENTROPY INEQUALITIES

9

´nyi entropy 2. Inequalities between dual mixed volumes and Re n Lemma 2.1. Suppose real p > 0 and λ > n+p . If K is a star body in Rn and X is a random vector in Rn that has finite pth moment, then ³ p p´ V˜−p (X, K) ≥ 1 + c0 [Nλ (X)/V (K)] n , n

with equality if and only if X is a random vector with density function a.e. proportional to pλ (a/ρK ), with some a > 0. Proof. Abbreviate

Z M= Rn

ρ−p f dx = K

n ˜ V−p (f, K). n+p

Let gλ,p = bpλ (a/ρK ), where b is chosen so that bpλ (a/ρK ) is a probability density and a is chosen so that Z ρ−p pλ (a/ρK ) dx. (2.1.1) M =b K Rn

First note that from Lemma 1.1 we know that Nλ (gλ,p ) ≥ Nλ (f ),

(2.1.2)

with equality if and only if a.e. gλ,p = f . From (2.1.1) and Lemma 0.1, we see that M = ba−(n+p) α1 V (K). From the fact that gλ,p is a density function and and the fact that gλ,p = bpλ (a/ρK ), together with Lemma 0.2 we see that Z 1= gλ,p dx = ba−n α2 V (K). Rn

Thus, we have: α1 α2 M an b= . α2 V (K)

ap =

(2.1.3) (2.1.4)

From Lemma 0.3, together with (2.1.3) and (2.1.4), we have Z Rn

λ gλ,p

λ −n

dx = b a

1 α3 V (K) = λ α2

µ

α1 α2 M

¶ n(λ−1) p

α3 V (K)1−λ .

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MOMENT-ENTROPY INEQUALITIES

Thus, for λ 6= 1,

µ Nλ (gλ,p )

p n

=

α3 α2λ

p ¶ n(1−λ)

p α2 M V (K) n . α1

(2.1.5)

Suppose λ = 1. From the definitions of g1,p , p1 , Ent1 , and the fact that bp1 (a/ρK ) is a probability density together with Lemma 0.1, (2.1.4), and finally (0.1.1) together with (0.1.2) we have: Z

p −p

be−a ρK (log b − ap ρ−p ) dx K Rn Z Z p ρ−p = − log b bp1 (a/ρK ) + a b p1 (a/ρK ) dx K

Ent1 (g1,p ) = −

Rn

Rn

−n

= − log b + ba α1 V (K) α1 = − log b + α2 n = − log b + . p This and the definition of N1 , together with (2.1.3) and (2.1.4), and finally (0.1.1) and (0.1.2) gives p p p ep p N1 (g1,p ) n = eb− n = α2n M V (K) n . (2.1.6) n Therefore, from (2.1.2) and (2.1.5), (or from (2.1.2) and (2.1.6) when λ = 1), we have M 1/p ≥ c00 [Nλ (f )/V (K)]1/n , with equality if and only if f = gλ,p , a.e., where c00 is given by

c00 =

 ³ ´ 1 ³ ´1 1   α3 n(λ−1) α1 p α− n 2 α2 α2

λ 6= 1

  ( n ) p1 α− n1 2 ep

λ = 1.

By using (0.1.1), (0.2.1), and (0.3.1) one now obtains the inequality of the lemma. ¤ A simple limit argument shows that the inequality of Lemma 2.1 holds when λ = ∞. However, in order to obtain the equality conditions we shall proceed in a different manner: Lemma 2.2. Suppose p > 0. If K is a star body in Rn and X is random vector in Rn that has finite pth moment and bounded density, then 1 1 V˜−p (X, K) p ≥ [N∞ (X)/V (K)] n ,

with equality if and only if there exists an a > 0 such that the density function of X is, a.e., 1aK /V (aK).

MOMENT-ENTROPY INEQUALITIES

11

Proof. Let f be the probability density of X, and let kf k∞ denote the essential supremum of f . We are assuming that kf k∞ < ∞, and for convenience let a = 1 [kf k∞ V (K)]− n . From the definition of a radial function (and the fact that K is a star body) it follows immediately that x ∈ int aK if and only if a > ρ−1 K (x) or equivalently int aK is −p p precisely the set on which the function (a − ρK )+ is positive. This observation and the fact that f is a density function, together with definition (0.6), shows that Z

Z p

(a − Rn

ρ−p K )+ f

dx ≥ Rn

(ap − ρ−p K )f dx

= ap −

n ˜ V−p (f, K), n+p

with equality if and only if f (x) = 0 for almost all x ∈ / int aK. Again from the fact that int aK is precisely the set on which the function (ap −ρ−p K )+ ˜ is positive, together with (0.2), the fact that V−p (·, K) is homogeneous of degree n + p, and (0.3) we have: Z

Z p

(a − Rn

ρ−p K )+ f

dx ≤ kf k∞

(ap − ρ−p K )+ dx

n

ZR = kf k∞

aK

(ap − ρ−p K ) dx

Z

= kf k∞ a

p

1 dx − aK

nkf k∞ ˜ V−p (aK, K) n+p

p = an+p kf k∞ V (K), n+p with equality if and only if f is a.e. constant on int aK. 1 Combining the above inequalities, and recalling that a = [kf k∞ V (K)]− n , gives n + p p p n+p V˜−p (f, K) ≥ a − a kf k∞ V (K) n n p = [ kf k∞ V (K) ]− n , with equality if and only if f is a.e. constant on int aK and 0 on its complement. This, and the fact that N∞ (f ) = 1/kf k∞ , gives the desired inequality and the fact that equality holds if and only if, a.e., f = 1aK /V (aK). ¤ 3. A star body associated with a random vector Suppose p > 0, and f : Rn → R is a density function that has finite pth moment. Define the Borel measure µf on S n−1 by letting Z S n−1

Z q(u) dµf (u) =

f (x)q(x/|x|)|x|p dx, Rn

12

MOMENT-ENTROPY INEQUALITIES

for each q ∈ C(S n−1 ). Since the measure µf is absolutely continuous with respect spherical Lebesgue measure, there is an essentially unique function f¯ ∈ L1 (S n−1 ) such that f¯ ≥ 0 and Z Z 1 ¯ q(u)f (u) du = f (x)q(x/|x|)|x|p dx, n S n−1 Rn for each q ∈ C(S n−1 ). Thus, from f we get a unique Ln+p -star, Tp f , defined by ρn+p = f¯ such that Tp f Z ˜ f (x)ρQ (x)−p dx, (3.1) V−p (Tp f, Q) = Rn

R n for each star body Q. Note that 0 < V (Tp f ) < ∞ (since V (Tp f ) = n1 S n−1 f¯n+p ), and 1 define Sp f = V (Tp f ) p Tp f . The homogeneity (of degree n) of volume now immediately 1 1 1 gives V (Sp f ) n+p = V (Tp f ) p . But V (Sp f )− n+p Sp f = Tp f and the fact that V˜−p (·, Q) is homogeneous of degree n + p, lets us rewrite (3.1) as Z ˜ f (x)ρQ (x)−p dx, (3.2) V−p (Sp f, Q)/V (Sp f ) = Rn

or by (0.2) equivalently n+p nV (Sp f )

Z

Z −p

Sp f

ρQ (x)

dx = Rn

f (x)ρQ (x)−p dx,

(3.3)

for each star body Q. If X random vector in Rn with density function f , that has finite pth moment, then we will often write Sp X rather than Sp f . Thus, from (3.2) and (0.6) we see that Sp X can be defined by simply requiring that ³ p´ ˜ ˜ V−p (Sp X, Q)/V (Sp X) = 1 + V−p (X, Q), (3.4) n hold for each star body Q. Observe that if f ∈ Cc∞ (Rn ) is positive in a neighborhood of the origin, then the Ln+p star, Sp f , is in fact a star body, and it follows that from (0.2) together with (3.2) that its radial function ρSp X is given by Z ∞ 1 n+p ρ (u) =n f (ru)rn+p−1 dr. V (Sp X) Sp X 0 If f ∈ Cc∞ (Rn ) is positive in a neighborhood of the origin, then by taking Q = Sp X in (3.4) and recalling (0.3) we see that p V˜−p (X, Sp X) = 1 + . n

(3.6)

As will now be shown, the volume of the star associated with a random vector can be bounded from below by the λ-R´enyi entropy power of the random vector. Although the inequality of our next lemma is stated without equality conditions, it is sharp.

MOMENT-ENTROPY INEQUALITIES

13

n Lemma 3.1. Suppose p > 0 and λ > n+p . If X is a random vector in Rn that has finite pth moment, then n/p V (Sp X) ≥ c0 Nλ (X).

Proof. Let f denote the density function of X. First note that if f ∈ Cc∞ (Rn ) is positive in a neighborhood of the origin, then from (3.6) and Lemma 2.1, ³ ³ n p ´ np p ´n/p n/p (3.1.1) 1+ = V˜−p (X, Sp X) p ≥ 1 + c0 Nλ (X)/V (Sp X). n n This establishes the desired inequality for the special case where f ∈ Cc∞ (Rn ) is positive in a neighborhood of the origin. To handle case for arbitrary f when λ 6= 1, choose a sequence of probability density functions fi ∈ Cc∞ (Rn ) that are positive in a neighborhood of the origin and such that Z Z −p f (x)ρQ (x)−p dx, (3.1.2) fi (x)ρQ (x) dx = lim i→∞

Rn

Rn

for each star body Q, and lim sup Nλ (fi ) ≥ Nλ (f ).

(3.1.3)

i→∞

Now suppose Q is a star body such that V (Q) = 1. From (3.1.2) and (3.2), followed by the dual mixed volume inequality (0.4), then (3.1.1), and finally (3.1.3), we have V˜−p (Sp f, Q)/V (Sp f ) = lim V˜−p (Sp fi , Q)/V (Sp fi ) i→∞

p

≥ lim sup V (Sp fi ) n i→∞

p

≥ c0 lim sup Nλ (fi ) n i→∞

p

≥ c0 Nλ (f ) n . This together with (0.5) now completes the proof for the case of arbitrary f when λ 6= 1. The case of arbitrary f when λ = 1 now follows from the case λ 6= 1 by taking a simple limit. ¤ It will be convenient to re-define c0 so that it is defined not only for positive λ but for λ = ∞ as well. To this end, define  ³ ´ 1 ³ ´ np ¡ ¢ n(1−λ) λ−1 pλ 1 n   1 − − 1 B np , 1−λ − np λ < 1,  p pλ n(1−λ)   ¡ ¢  n   ( pe ) p Γ n + 1 λ = 1, n p −n/p c0 = ³ ´ 1 ³ ´ np ¡ ¢  n(λ−1) λ−1  pλ n λ  1 + + 1 B np , λ−1 λ > 1,  p pλ n(λ−1)    ¡ ¢n λ = ∞. 1 + np p Taking λ → ∞ in Lemma 3.1 (and noting that limλ→∞ c0 = (1 + p/n)−1 ) gives: Lemma 3.2. Suppose real p > 0 and X is a random vector in Rn that has finite pth moment, then n/p V (Sp X) ≥ c0 N∞ (X).

14

MOMENT-ENTROPY INEQUALITIES

4. Moments of random variables Suppose p ≥ 1. If K ⊂ Rn is an Ln+p -star that is of positive volume, then its polar Lp -centroid body, Γ∗p K, is the convex body whose radial function is given by −p

ρΓ∗p K (u)

1 = V (K)

Z |u · x|p dx.

(4.1)

K

From this definition, it is easily seen that if E is an ellipsoid centered at the origin then Γ∗p E is a dilate of E ∗ . (4.2) Definitions (0.2) and (4.1) and Fubini’s theorem show that for positive-volume Ln+p stars K, L ⊂ Rn we have V˜−p (K, Γ∗p L)/V (K) = V˜−p (L, Γ∗p K)/V (L).

(4.3)

Take L = Γ∗p K in (4.3), and from (0.3) get: If K ⊂ Rn is an Ln+p -star, then V (K) = V˜−p (K, Γ∗p Γ∗p K).

(4.4)

For p = 1 and a more restricted class of bodies identity (4.3) was first given in Lutwak (1990). Lemma 4.2. Let X and Y be independent random variables in Rn that have finite pth moment, then for p ≥ 1 V˜−p (X, Γ∗p Sp Y ) = E(|X · Y |p ) = V˜−p (Y, Γ∗p Sp X). Proof. Let f, g be the density functions of X, Y . First note that an obvious limit argument in (3.3) shows that for each x ∈ Rn , Z Z n+p p |x · y| dy = g(y)|x · y|p dy. (4.2.1) nV (Sp g) Sp g n R By using (0.6), (4.1), and (4.2.1), we have Z p ∗ ˜ V−p (X, Γp Sp Y ) = (1 + n )

Rn

ρ−p Γ∗ S

(x)f (x)dx Z f (x) |x · y|p dy dx

p pY

Z n+p = nV (Sp Y ) Rn Sp Y Z Z = |x · y|p f (x)g(y) dx dy Rn

Rn

= E(|X · Y |p ). To complete the proof observe that E(|X · Y |p ) is symmetric in X and Y .

¤

MOMENT-ENTROPY INEQUALITIES

Define c2 by

à c2 =

15

¡ ¢ ¡ ¢ ! np 1 π 2 +p 1 + np Γ p+n 2 . ¡ n ¢ ¡ p+1 ¢ ¡ ¢2p/n Γ 2 Γ 2 Γ 1 + n2

We shall use the following result of Lutwak and Zhang (1997) (see also Lutwak, Yang, and Zhang (2000) as well as Campi and Gronchi (2002)): If K is an origin-symmetric convex body, then V (Γ∗p K)V (K) ≤ c2 , (4.5) with equality if and only if K is an ellipsoid centered at the origin. n Theorem 4.3. Suppose real p ≥ 1 and λ ∈ ( n+p , ∞]. Let X and Y be independent n th random variables in R that have finite p moment (and are also of bounded density if λ = ∞), then −p

p

E(|X · Y |p ) ≥ c20 c2 n (1 + np )[Nλ (X)Nλ (Y )] n , with equality for λ < ∞ if and only if there exists an φ ∈ GL(n) with X having density a.e. φp,λ and Y having density a.e. (aφ−t )p,λ , with a > 0, and equality for λ = ∞ if and only if the densities of X and Y are a.e. proportional to the characteristic functions of dilates of polar-reciprocal origin-centered ellipsoids. Proof. With i 6= j, let {Xi , Xj } = {X, Y } and let fi , fj denote the density functions of Xi , Xj . From Lemma 4.2, (3.4), the dual mixed volume inequality (0.4), Lemma 3.2, (4.5), (0.4), (4.4), and Lemma 3.2, we have E(|X · Y |p ) = V˜−p (Xi , Γ∗p Sp Xj ) = (1 + p )V˜−p (Sp Xi , Γ∗p Sp Xj )/V (Sp Xi ) ≥ (1 + ≥ (1 + ≥ (1 +

n p p ∗ n n )[V (Sp Xi )/V (Γp Sp Xj )] p p ∗ n n )c0 [Nλ (Xi )/V (Γp Sp Xj )] p p p −n p ∗ ∗ n n )c c 0 2 Nλ (Xi ) V (Γp Γp Sp Xj ) n p

n+p p − ≥ (1 + np )c0 c2 n Nλ (Xi ) n V˜−p (Sp Xj , Γ∗p Γ∗p Sp Xj )−1 V (Sp Xj ) n

−p

p

p

= (1 + np )c0 c2 n Nλ (Xi ) n V (Sp Xj ) n −p

p

p

≥ (1 + np )c20 pc2 n Nλ (Xi ) n Nλ (Xj ) n . Suppose there is equality in the inequality of the theorem. Obviously, this implies equality in all of the inequalities above. Equality in the first inequality, by the equality conditions of the dual Minkowski inequality (0.4), implies that there exist di > 0 such that, a.e., ρSp Xi = di ρΓ∗ Sp Xj . (4.3.1) p

16

MOMENT-ENTROPY INEQUALITIES

p The fact that V˜−p (Xi , Γ∗p Sp Xj ) = (1 + np )c0 [Nλ (Xi )/V (Γ∗p Sp Xj )] n and the equality conditions of Lemma 2.1 (or Lemma 2.2 if λ = ∞) shows that there exist ai , bi > 0 such that, a.e., ( λ < ∞, bi pλ (ai /ρΓ∗ Sp Xj ) p fi = (4.3.2) 1ai Γ∗p Sp Xj /V (ai Γ∗p Sp Xj ) λ = ∞.

From the equality conditions of (4.5), we see that equality in the third inequality implies, that there exits an origin-centered ellipsoid, Ej , such that Γ∗p Sp Xj = Ej . But (4.3.3) and (4.3.2) show that a.e., ½ bi pλ (ai /ρEj ) fi = 1ai Ej /V (ai Ej )

(4.3.3)

λ < ∞, λ = ∞.

(4.3.4)

But (4.3.1) and (4.3.3) show that ρSp Xi = di ρEj , a.e., and hence by (4.1) Γ∗p Sp Xi = d0i Ej∗ ,

(4.3.5)

for some d0i > 0. Note that (4.3.2) shows that, a.e., ( bj pλ (aj /ρΓ∗ Sp Xi ) p fj = 1aj Γ∗p Sp Xi /V (aj Γ∗p Sp Xi ) and when combined with (4.3.5) we see that, a.e., ( bj pλ (aj /ρd0 E∗ ) i j fj = 0 ∗ 1aj Ej /V (a0j Ej∗ )

λ < ∞, λ = ∞,

λ < ∞, λ = ∞,

(4.3.6)

for some a0j > 0. When (4.3.4) and (4.3.6) are combined with (0.1), we get the desired equality conditions. ¤ A comment regarding the above proof: Two extra steps were needed in the proof of the inequality of Theorem 4.3 because inequality (4.5) had not been established for the class of Ln+p -stars. The class reduction technique that is used to overcome this difficulty was introduced in Lutwak (1986). Suppose K, L ⊂ Rn are compact. In Theorem 4.3, with λ = ∞, let X and Y be random vectors with density 1K /V (K) and 1L /V (L) and get: Corollary 4.4. If K and L are compact sets in Rn , then for real p ≥ 1, Z Z ¤ n+p n − np £ |x · y|p dxdy ≥ c2 V (K)V (L) n , n+p K L with equality if and only if K and L are, up to sets of measure 0, dilates of polarreciprocal, origin-centered ellipsoids. The limiting case p → ∞ of Corollary 4.4 gives:

MOMENT-ENTROPY INEQUALITIES

17

Corollary 4.5. If K and L are compact sets in Rn , then −2

1

max |x · y| ≥ ωn n [V (K)V (L)] n .

x∈K,y∈L

If K is an origin-symmetric convex body and L is its polar, then the above inequality −2/n is the classical Blaschke-Santal´ o inequality (and ωn is the precise constant that makes the inequality sharp). References Bourgain, J. and Milman V. D. (1987), New volume ratio properties for convex symmetric bodies in Rn , Invent. Math. 88, 319–340. Campi, S. and Gronchi, P. (2002), The Lp Busemann-Petty centroid inequality, Advances in Math. 167, 128–141. Cover, C. M. and Thomas, J. A. (1992), Elements of information theory, Wiley, New York. Gardner, R. J. (1995), Geometric Tomography, Cambridge Univ. Press, Cambridge. Gardner, R. J. (2002), The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39, 355–405. Gardner, R. J. and Volˇ ciˇ c, A. (1994), Tomography of convex and star bodies, Advances in Math. 108, 367–399. Gardner, R. J., Vedel Jensen, E. B. and Volˇ ciˇ c, A., Geometric tomography and local stereology, Advances in Appl. Math. 30 (2003), 397–423. Klain, D. A. (1996), Star valuations and dual mixed volumes, Adv. Math 121, 80–101. Klain, D. A. (1997), Invariant valuations on star-shaped sets, Adv. Math 125, 95–113. Leichtweiß, K. (1998), Affine geometry of convex bodies, Johann Ambrosius Barth, Heidelberg. Lutwak, E. (1986), On some affine isoperimetric inequalities, J. Differential Geom. 23, 1–13. Lutwak, E. (1990), Centroid bodies and dual mixed volumes, Proc. London Math. Soc. 60, 365–391. Lutwak, E., Yang, D. and Zhang, G. (2000), Lp Affine Isoperimetric Inequalities, J. Differential Geom. 56, 111-132. Lutwak, E. and Zhang, G. (1997), Blaschke-Santal´ o inequalities, J. Differential Geom. 47, 1–16. Schneider, R. (1993), Convex Bodies: the Brunn–Minkowski Theory, Cambridge Univ. Press, Cambridge. Thompson, A. C. (1996), Minkowski Geometry, Cambridge Univ. Press, Cambridge. Vitale, R. A. (1996a), The Wills functional and Gaussian processes, Ann. Probab. 24, 2172-2178. Vitale, R. A. (1996b), Covariance identities for normal variables via convex polytopes, Statist. Probab. Lett. 30, 363–368. Vitale, R. A. (2001), Intrinsic volumes and Gaussian processes, Adv. in Appl. Probab. 33, 354–364.

Erwin Lutwak, Department of Mathematics, Polytechnic University, Brooklyn, N.Y. 11201, [email protected] Deane Yang, Department of Mathematics, Polytechnic University, Brooklyn, N.Y. 11201, [email protected] Gaoyong Zhang, Department of Mathematics, Polytechnic University, Brooklyn, N.Y. 11201, [email protected]

Moment-entropy inequalities

MOMENT-ENTROPY INEQUALITIES. Erwin Lutwak, Deane Yang and Gaoyong Zhang. Department of Mathematics. Polytechnic University. Brooklyn, NY 11201.

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