Divergence for s-concave and log concave functions
∗
Umut Caglar and Elisabeth M. Werner†
Abstract
We prove new entropy inequalities for log concave and s-concave functions that
strengthen and generalize recently established reverse log Sobolev and Poincaré
inequalities for such functions. This leads naturally to the concept of f -divergence
and, in particular, relative entropy for s-concave and log concave functions. We
establish their basic properties, among them the affine invariant valuation property.
Applications are given in the theory of convex bodies.
1
Introduction
There is a general approach to extend invariants and inequalities of convex bodies to
the corresponding invariants and inequalities for functions. Among the best known
affine isoperimetric inequalities is the Blaschke Santaló inequality [10, 49, 59]. The
corresponding inequalities for log concave functions were proved by Ball [7] and Artstein,
Klartag and Milman [5] (see also [17, 31]). A stronger inequality than the Blaschke
Santaló inequality is the affine isoperimetric inequality for convex bodies [10, 15, 59].
The equivalent of this inequality for log concave functions was established in [6]: For
n
every log-concave function
R ϕ : R → [0, ∞) with enough smoothness and integrability
properties and such that ϕdx = 1,
Z
ϕ ln det (Hess (− ln ϕ)) dx ≤ 2 [Ent(ϕ) − Ent(g)] ,
(1)
supp(ϕ)
2 ϕ
where g is the Gaussian, supp(ϕ) is the support, Hess(ϕ) = ∂x∂i ∂x
is the
j
1≤i,j≤n
R
Hessian and Ent(ϕ) = supp(ϕ) ϕ ln ϕdx is the entropy of ϕ. Thus, the affine isoperimetric
inequality corresponds to a reverse log Sobolev inequality for entropy. Equality holds in
(1) if and only if ϕ(x) = Ce−hAx,xi , where C is a positive constant and A is an n × n
positive definite matrix. This characterization of equality in inequality (1) was achieved
in [12].
Here, we strengthen and generalize inequality (1).
Inequality (1) is yet another instance of the rapidly developing, fascinating connection
between convex geometric analysis and information theory. Further examples can be
found in e.g., [16, 28, 42, 44, 46, 47, 54]. In particular, it has been observed [67] that a
∗ Keywords: entropy, divergence, affine isoperimetric inequalities, log Sobolev inequalities. 2010
Mathematics Subject Classification: 46B, 52A20, 60B
† Partially supported by an NSF grant
1
fundamental notion of affine convex geometry, the Lp -affine surface area can be viewed
as Rényi entropy from information theory, thus establishing a link between information
theory and the powerful Lp -Brunn-Minkowski theory [41] of affine convex geometry.
Due to a number of highly influential works (see, e.g., [19]- [23], [29], [30], [34]- [48],
[58], [62]- [67], [69], [72]), this theory is now a central part of modern convex geometry.
Rényi entropies are special cases of f -divergences whose definition is given in Section 2.
Such divergences and their related inequalities are important tools in information theory,
statistics, probability theory and machine learning [8, 13, 18, 24, 32, 33, 53, 55, 71].
Consequently, it is desirable to have such divergences available also in the theory of
convex bodies and this was achieved in [68].
In this paper, we further develop that direction. We introduce f -divergences for
functions and establish some of their basic properties, among them the affine invariance
property and the valuation property. Valuations were the critical ingredient in Dehn’s
solution of Hilbert’s third problem and, in the last decade, have seen rapid growth as is
demonstrated by e.g., [1]-[3], [9], [22], [34]- [40], [60].
We prove the following entropy inequality for log concave functions, i.e. functions of
the form ϕ = e−ψ with ψ : Rn → R convex. This inequality is stronger than inequality
(1). Its proof uses methods different from the ones used in [6].
Theorem 1. Let f : (0, ∞) → R be a convex function. Let ϕ : Rn → [0, ∞) be a
log-concave function. Then
Z
∇ϕ
ϕ f eh ϕ ,xi ϕ−2 (det (Hess (− ln ϕ))) dx
supp(ϕ)
!
R ◦ Z
ϕ dx
≥ f R
ϕdx .
(2)
ϕdx
supp(ϕ)
If f is concave, the inequality is reversed. If f is linear, equality holds in (2). Equality
also holds if ϕ(x) = Ce−hAx,xi , where C is a positive constant and A is an n × n positive
definite matrix.
Here, ∇ϕ denotes the gradient of ϕ and ϕ◦ = inf y∈Rn
h
e−hx,yi
ϕ(y)
i
[5] is the dual function
of ϕ. We will demonstrate that the left hand side of the inequality (2) is the natural
definition of f -divergence Df (ϕ) for a log concave function ϕ, so that inequality (2) can
be rewritten as
!
R ◦ Z
ϕ dx
Df (ϕ) ≥ f R
ϕdx .
(3)
ϕdx
supp(ϕ)
This is shown in Section 3. Inequality (3) also holds for s-concave functions. We prove
this in Theorem 8.
If we let f (t) = ln t in Theorem 1, we obtain the following corollary.
Corollary 2. Let ϕ : Rn → [0, ∞) be a log-concave function. Then
Z Z
Z
n
◦
ϕ ln det (Hess (− ln ϕ)) dx ≤ 2 Ent(ϕ) + kϕkL1 ln e
ϕ
ϕ
, (4)
supp(ϕ)
with equality if ϕ(x) = Ce−hAx,xi , where C is a positive constant and A is an n × n
positive definite matrix.
2
We show in Section 3 that inequality (4) involves the relative entropy or Kullback
Leibler divergence DKL (ϕ) (see Section 2 for the definition) of the function ϕ and thus
inequality (4) is equivalent to
! R
Z
ϕ◦ dx
.
DKL (ϕ) ≤
ϕdx ln R
ϕdx
supp(ϕ)
Moreover, as it is shown in Section 3, the inequality of Corollary 2 is stronger than
inequality (1).
It is important to note the affine invariant nature of the expressions (2), (4) and of
(5) below. Both, the respective left-hand sides and the right-hand sides, are invariant
under volume-preserving linear transformations.
The key ingredient to prove Theorem 1 is (a special case) of the following duality
relation for log concave functions ϕ : Rn → [0, ∞) and their duals ϕ◦ .
Theorem 3. Let ϕ : Rn → [0, ∞) be a log-concave function. For a convex or concave
function f : (0, ∞) → R, let f ∗ (t) = tf (1/t). Then
Df (ϕ◦ ) = Df ∗ (ϕ).
(5)
We present several applications. In Section 4, we consider f -divergence for special
functions f , which, on the level of convex bodies, correspond to Lp -affine surface areas.
We refer to [41, 51, 63] for the definition and to e.g., [23], [37], [39], [40], [50], [61], [62],
[66]-[70] for more information on Lp -affine surface area for convex bodies. The Lp -affine
surface areas for functions were already introduced in [12]. Here, we establish several
affine isoperimetric inequalities for these quantities. They are the functional counterparts
of known inequalities for convex bodies. Another application is given in Section 5, where
we apply our results about log concave functions to convex bodies. Finally, in Section 6
we obtain a reverse Poincaré inequality that is stronger than the one proved in [6].
Throughout the paper we will assume that the convex or concave functions f :
(0, ∞) → R and the s-concave and log concave functions ϕ : Rn → [0, ∞) have enough
smoothness and integrability properties so that the expressions considered in the statements make sense, i.e., we will always assume that ϕ◦ ∈ L1 (supp(ϕ), dx), the Lebesgue
integrable functions on the support of ϕ, that
ϕ ∈ C 2 (supp(ϕ)) ∩ L1 (Rn , dx),
(6)
where C 2 (supp(ϕ)) denotes the twice continuously differentiable functions on their support, and that
!
h∇ϕ,xi
e ϕ
ϕf
det (Hess (− ln ϕ)) ∈ L1 (supp(ϕ), dx).
(7)
ϕ2
See also Remark (iv) after Definition 9.
3
2
f -divergence for s-concave functions.
2.1
Background on f -divergence.
In information theory, probability theory and statistics, an f -divergence is a function
that measures the difference between two (probability) distributions. This notion was
introduced by Csiszár [14], and independently Morimoto [52] and Ali & Silvery [4].
Let (X, µ) be a measure space and let P = pµ and Q = qµ be (probability) measures
on X that are absolutely continuous with respect to the measure µ. Let f : (0, ∞) → R
be a convex or a concave function. The ∗-adjoint function f ∗ : (0, ∞) → R of f is defined
by
f ∗ (t) = tf (1/t), t ∈ (0, ∞).
(8)
It is obvious that (f ∗ )∗ = f and that f ∗ is again convex if f is convex, respectively
concave if f is concave. Then the f -divergence Df (P, Q) of the measures P and Q is
defined by
Z
p
Df (P, Q) =
f
qdµ + f (0) Q ({x ∈ X : p(x) = 0})
q
{pq>0}
+ f ∗ (0) P ({x ∈ X : q(x) = 0}) ,
(9)
provided the expressions exist. Here
and f ∗ (0) = lim f ∗ (t).
f (0) = lim f (t)
t↓0
t↓0
(10)
We make the convention that 0 · ∞ = 0.
Please note that
Df (P, Q) = Df ∗ (Q, P ).
(11)
With (10) and as
f ∗ (0) P ({x ∈ X : q(x) = 0}) =
Z
f∗
{q=0}
we can write in short
Z
Df (P, Q) =
f
X
Z
q
p
pdµ =
f
qdµ,
p
q
{q=0}
p
qdµ.
q
(12)
Examples of f -divergences are as follows.
1. For f (t) = t ln t (with ∗-adjoint function f ∗ (t) = − ln t), the f -divergence is KullbackLeibler divergence or relative entropy from P to Q (see [13])
Z
p
DKL (P kQ) =
p ln dµ.
(13)
q
X
2. For the convex or concave functions f (t) = tα we obtain the Hellinger integrals (e.g.
[33])
Z
Hα (P, Q) =
pα q 1−α dµ.
(14)
X
4
Those are related to the Rényi divergence of order α, α 6= 1, introduced by Rényi [56]
(for α > 0) as
Z
1
1
α 1−α
ln
p q
dµ =
ln (Hα (P, Q)) .
(15)
Dα (P kQ) =
α−1
α
−
1
X
The case α = 1 is the relative entropy DKL (P kQ).
More on f -divergence can be found in e.g. [18, 32, 33, 53, 55, 68, 71].
2.2
f -divergence for s-concave functions.
Let s ∈ R, s 6= 0. Let ϕ : Rn → R+ . Following Borell [11], we say that ϕ is s-concave if
for every λ ∈ [0, 1] and all x and y such that ϕ(x) > 0 and ϕ(y) > 0,
ϕ((1 − λ)x + λy) ≥ ((1 − λ)ϕ(x)s + λϕ(y)s )
1/s
.
Note that s can be negative. Now we want to define f -divergence for s-concave functions.
To do that, let
h
i
det −Hess(ϕ)
+ (1 − s) ∇ϕ⊗∇ϕ
2
ϕ
ϕ
h∇ϕ, xi
(s)
(s)
Pϕ =
, Qϕ = ϕ 1 − s
.
n+ 1s
ϕ
ϕ 1 − s h∇ϕ,xi
ϕ
Recall that we assume that the functions satisfy the conditions (6) and (7).
Definition 4. Let f : (0, ∞) → R be a convex or concave function. Let
let
s ∈ R and
(s)
(s)
(s)
n
ϕ : R → [0, ∞) be an s-concave function. Then the f -divergence Df Pϕ , Qϕ of ϕ
is

h
i
−Hess(ϕ)
∇ϕ⊗∇ϕ
Z
det
+
(1
−
s)
2
ϕ
ϕ
h∇ϕ, xi


(s)
(s)
(s)
ϕ f
Df Pϕ , Qϕ =
dx.
 1−s
n+ 1s +1
ϕ
supp(ϕ)
ϕ2 1 − s h∇ϕ,xi
ϕ
(s)
(s)
We will sometimes write in short Df (ϕ) for Df
(s)
(s)
Pϕ , Qϕ
.
Please note also that for s 6= 1 expression of the definition can be rewritten as
(s)
(s)
Df (ϕ) = Df (Pϕ(s) , Q(s)
ϕ )=

h
i 
∇ϕ⊗∇ϕ
Z
det
Hess
(−
ln
ϕ)
+
s
2
ϕ
h∇ϕ, xi


ϕf 
1
−
s
dx.

n+ 1s +1
ϕ
h∇ϕ,xi
supp(ϕ)
2
ϕ 1−s ϕ
(s)
(s)
(s)
(16)
Remark. A similar expression holds for Df (Qϕ , Pϕ ), namely
(s)
(s)
Df Q(s)
=
ϕ , Pϕ


n+ 1s +1
h
i
h∇ϕ,xi
∇ϕ⊗∇ϕ
2
Z
ϕ
1
−
s
det
Hess
(−
ln
ϕ)
+
s
2
ϕ
ϕ

h
i 
f
.(17)

n+ 1s
∇ϕ⊗∇ϕ
h∇ϕ,xi
supp(ϕ)
det Hess (− ln ϕ) + s ϕ2
ϕ 1−s
ϕ
5
(s)
(s)
By (11), Df
consider Df
(s)
(s)
(s)
(s)
Qϕ , Pϕ
Pϕ , Qϕ
(s)
= Df ∗
(s)
(s)
Pϕ , Qϕ
(s)
= Df ∗ (ϕ). Therefore it is enough to only
. We will do this throughout the paper.
The motivation for this definition of f -divergence for s-concave functions comes from
convex geometry. In [68], f -divergence for a convex body K in Rn was introduced. We
refer to [68] for more information and special cases and give here only the definition.
For x ∈ ∂K, the boundary of a sufficiently smooth convex body K, let NK (x) denote
the outer unit normal to ∂K in x and let κK (x) be the Gauss curvature in x. µK is the
usual surface area measure on ∂K. We put
pK (x) =
κK (x)
, qK (x) = hx, NK (x)i
hx, NK (x)in
(18)
and
PK = pK µK
and QK = qK µK .
(19)
Then PK and QK are measures on ∂K that are absolutely continuous with respect to
µK . Let f : (0, ∞) → R be a convex or concave function. In fact, QK and PK are (up
to the factor n) the cone measures (e.g. [54]) of K and its polar
K ◦ = {y : hx, yi ≤ 1 ∀x ∈ K},
(20)
the latter provided K has sufficiently smooth boundary.
The f -divergence of K with respect to the measures PK and QK was defined in [68]
as
Z
Df (PK , QK ) =
f
∂K
pK
qK
Z
qK dµK =
f
∂K
κK (x)
hx, NK (x)in+1
hx, NK (x)idµK . (21)
For s > 0 such that 1s ∈ N, we associate with an s-concave function ϕ a convex body
1
Ks (ϕ) [5] (see also [6]) in Rn × R s ,
p
p
1
Ks (ϕ) = (x, y) ∈ Rn × R s : 1/s x ∈ supp(ϕ), kyk ≤ ϕs ( 1/s x) .
(22)
The following proposition relates the definitions of f -divergence for the convex bodies
and s-concave functions.
Proposition 5. Let s > 0 be such that 1s ∈ N. Let f : (0, ∞) → R be a convex or
concave function. Let ϕ : Rn → [0, ∞) be an s-concave function. Then
D P
f
Ks (ϕ) , QKs (ϕ)
(s)
(s)
(s)
Df Pϕ , Qϕ =
,
(23)
n
1
s 2 vol 1s −1 S s −1
1
where S s −1 is the
1
s
− 1 -dimensional Euclidean sphere.
Proof. It was shown in [6] that for z ∈ ∂(Ks (ϕ))
hz, NKs (ϕ) (z)i =
ϕs − h∇ (ϕs ) , xi
1
(1 + k∇ (ϕs ) k2 ) 2
6
(24)
and
1
hz, NKs (ϕ) (z)in+ s +1
+ (1 − s) ∇ϕ⊗∇ϕ
2
ϕ
n+ 1s +1 ,
√
h∇ϕ,xi
ϕ2 1 − s ϕ
det
κKs (ϕ) (z)
=
−Hess ϕ
ϕ
(25)
p
p
p
n
where ϕ is evaluated at 1/sx = ( 1/sx1 , . . . , 1/sx
We denote the collection
p n) ∈ R . p
of all points (x1 , . . . , xn+ 1s ) ∈ ∂Ks (ϕ) such that ( 1/sx1 . . . , 1/sxn ) ∈ int(supp(ϕ))
˜ s (ϕ). Since there is no contribution to the integral of Df PK (ϕ) , QK (ϕ) from
by ∂K
s
s
˜ s (ϕ) (since the Gauss curvature vanishes on the part with full dimension, if
∂Ks (ϕ) \ ∂K
it exists), we get with (24) and (25),
Z
κKs (ϕ) (z)
hz, NKs (ϕ) (z)idµKs (ϕ)
Df (PKs (ϕ) , QKs (ϕ) ) =
f
hz, NKs (ϕ) (z)in+1
∂Ks (ϕ)


−Hess ϕ
∇ϕ⊗∇ϕ
Z
det
+
(1
−
s)
2
ϕ
ϕ

 (ϕs − h∇ (ϕs ) , xi)
=
f

1 dµKs (ϕ)
n+s+1
√ h∇ϕ,xi
s ) k2 ) 2
˜ s (ϕ)
∂K
2
(1
+
k∇
(ϕ
ϕ
1− s ϕ


−Hessϕ
∇ϕ⊗∇ϕ
Z
det
+
(1
−
s)
ϕ
ϕ2
(ϕs ) , xi

 ϕs − h∇
dx1 . . . dxn+ 1 −1
=2
f


n+ 1s +1
1
s
√
Rn+ s −1
ϕ−s xn+ 1s ϕ2 1 − s h∇ϕ,xi
ϕ
p
p
p
where ϕ is evaluated at 1/sx = ( 1/sx1 , . . . , 1/sxn ). The last equality follows as
the boundary of Ks (ϕ) consists of two, “positive” and “negative”, parts. As in [6],
p
Z
1
dxn+1 . . . dxn+ 1s −1
ϕ1−2s ( 1/sx)
1
vol
=
B2s .
1
s
2s
R s −1
1
xn+ s Hence,
Df (PKs (ϕ) , QKs (ϕ) ) =


−Hess ϕ
∇ϕ⊗∇ϕ
det
+
(1
−
s)
√ h∇ϕ, xi
ϕ
ϕ2


dx,
n √
o ϕf 
n+ 1s +1  1 − s ϕ
√
x: 1/sx∈supp(ϕ)
ϕ2 1 − s h∇ϕ,xi
ϕ
Z
cs
p
p
p
where ϕ is evaluated at 1/sx = ( 1/sx1 , . . . , 1/sxn ) and where cs =
1 p
vol 1s −1 S s −1 . With the change of variable 1/sx = y,
1
s
1
vol 1s B2s =
Df (PKs (ϕ) , QKs (ϕ) ) =

h i 
Hess(ϕ)
∇ϕ⊗∇ϕ
Z
det
−
−
(1
−
s)
ϕ
ϕ2
n
h∇ϕ, yi


2
cs s
ϕf
dy =
 1−s
n+ 1s +1
ϕ
supp(ϕ)
ϕ2 1 − s h∇ϕ,yi
ϕ
n
(s)
cs s 2 Df Pϕ(s) , Q(s)
.
ϕ
7
Now we describe some properties of the f -divergence for s-concave functions. By
(s)
(s)
(s)
(11), it is enough to do this for Df (Pϕ , Qϕ ) only.
Lemma 6. Let ϕ : Rn → [0, ∞) be an s-concave function and let f : (0, ∞) → R be
(s)
(s)
(s)
(s)
a convex or concave function. Then Df (ϕ) = Df (Pϕ , Qϕ ) is invariant under self
adjoint SL(n) invariant linear maps and it is a valuation: If max(ϕ1 , ϕ2 ) is s-concave,
then
(s)
(s)
(s)
(s)
Df (ϕ1 ) + Df (ϕ2 ) = Df (max(ϕ1 , ϕ2 )) + Df (min(ϕ1 , ϕ2 )).
Proof. Let A : Rn → Rn be a self adjoint, SL(n) invariant linear map. By Definition 4,
Z
h∇ (ϕ(Ax)) , xi
(s)
(s)
(s)
Df (Pϕ◦A , Qϕ◦A ) =
ϕ(Ax) 1 − s
ϕ(Ax)
supp(ϕ◦A)


−ϕ(Ax) Hess(ϕ(Ax))+(1−s)(∇(ϕ(Ax))⊗∇(ϕ(Ax)))
det
(ϕ(Ax))2


f
 dx
n+ 1s +1
h∇(ϕ(Ax)),xi
(ϕ(Ax))2 1 − s
ϕ(Ax)

h
i
−Hess(ϕ)
∇ϕ⊗∇ϕ
Z
det
+
(1
−
s)
ϕ
ϕ2
h∇ϕ, yi
1


ϕ 1−s
f (detA)2
=
 dy
1
n+
+1
|detA| supp(ϕ)
ϕ
s
ϕ2 1 − s h∇ϕ,yi
ϕ
(s)
= Df (Pϕ(s) , Q(s)
ϕ ).
Next. we establish the valuation property. There, AC denotes the complement of a
set A ⊂ Rn .
(s)
(s)
(s)
(s)
(s)
Df Pϕ(s)
,
Q
ϕ1 + Df (Pϕ2 , Qϕ2 ) =
1

h
i
−Hess(ϕ1 )
∇ϕ1 ⊗∇ϕ1
Z
+
(1
−
s)
det
2
ϕ1
h∇ϕ1 , xi
ϕ1


ϕ1 f 
dx +
 1−s
n+ 1s +1
ϕ1
supp(ϕ1 )∩supp(ϕ2 )
ϕ21 1 − s h∇ϕϕ11,xi

h
i
−Hess(ϕ1 )
∇ϕ1 ⊗∇ϕ1
Z
+
(1
−
s)
det
2
ϕ1
h∇ϕ1 , xi
ϕ1


1
−
s
dx +
ϕ1 f 

n+ 1s +1
ϕ1
h∇ϕ1 ,xi
supp(ϕ1 )∩(supp(ϕ2 ))C
2
ϕ1 1 − s ϕ1

h
i
−Hess(ϕ2 )
∇ϕ2 ⊗∇ϕ2
Z
det
+
(1
−
s)
2
ϕ2
h∇ϕ2 , xi
ϕ2


ϕ2 f 
1
−
s
dx +

n+ 1s +1
ϕ2
h∇ϕ2 ,xi
supp(ϕ1 )∩supp(ϕ2 )
2
ϕ2 1 − s ϕ2

i
h
−Hess(ϕ2 )
∇ϕ2 ⊗∇ϕ2
Z
det
+
(1
−
s)
2
ϕ2
h∇ϕ2 , xi
ϕ2


1
−
s
dx
ϕ2 f 

n+ 1s +1
ϕ2
h∇ϕ2 ,xi
supp(ϕ2 )∩(supp(ϕ1 ))C
2
ϕ2 1 − s ϕ2
(s)
(s)
(s)
(s)
(s)
(s)
= Df (Pmax(ϕ1 ,ϕ2 ) , Qmax(ϕ1 ,ϕ2 ) ) + Df (Pmin(ϕ1 ,ϕ2 ) , Qmin(ϕ1 ,ϕ2 ) ),
provided that max(ϕ1 , ϕ2 ) is s-concave.
8
Let s ∈ R, s 6= 0 and let ϕ : Rn → R+ be an s-concave function. Let supp(ϕ) = {x :
ϕ(x) > 0} be the support of ϕ. Then supp(ϕ) is convex. We will assume throughout the
rest of this section that supp(ϕ) is open and bounded, that ϕ is C 2 on supp(ϕ) and that
limx→∂ supp(ϕ) ϕs (x) = 0. We define a function ψ on supp(ϕ) (see also [57]) by
ψ(x) =
1 − ϕs (x)
, x ∈ supp(ϕ).
s
(26)
As ϕ > 0 on supp(ϕ), ψ is well defined, ψ is convex on supp(ϕ), ψ < 1s , if s > 0
and ψ > 1s , if s < 0. We will use the following duality definition from [12]. First,
∗
let (supp(ϕ)) = {y : supx∈supp(ϕ) hx, yi < 1}. Note that we can assume without
loss of generality that 0 ∈ supp(ϕ). If 0 ∈
/ supp(ϕ), pick z ∈ supp(ϕ) and consider
∗
∗
∗
(supp(ϕ) − z) + z. Then (supp(ϕ)) is convex, open, bounded and 0 ∈ (supp(ϕ)) . On
(supp(ϕ))∗
we define
the set
s
∗
?
ψ(s)
(y) =
(supp(ϕ))
hx, yi − ψ(x)
, y∈
.
sup
1
−
sψ(x)
s
x∈supp(ϕ)
(27)
∗
?
Then ψ(s)
is convex, and, as for s > 0, hx, yi < 1s for x ∈ supp(ϕ) and y ∈ (supp(ϕ)) ,
?
?
> 1s , if s < 0. Observe also that
we have that ψ(s)
< 1s , if s > 0 and, similarly, that ψ(s)
for s → 0 we obtain the Legendre transform Lψ(y) = supx [hx, yi − ψ(x)]. We denote
1/s
∗
ϕ∗(s) (x) = 1 − sψ(s)
(x)
∗
the function corresponding to ψ(s)
. ϕ∗(s) is well defined, s-concave and, putting ϕ∗(s) ≡ 0
1/s
(1−shx,yi)+
(supp(ϕ))∗
outside
,
coincides
for
s
>
0
with
L
(ϕ)(y)
=
inf
from [5].
s
supp(ϕ)
s
ϕ(x)
The supremum in (27) is attained at x such that
y=
1 − shx, yi
?
∇ψ(x) which means y = (1 − sψ(s)
(y))∇ψ(x).
1 − sψ(x)
Moreover,
1
1 − sψ(x)
? (y) = 1 − shx, yi = 1 + s(h∇ψ(x), xi − ψ(x)),
1 − sψ(s)
(28)
and the relation between y and x is
y=
∇ψ(x)
= Tψ (x).
1 + s(h∇ψ(x), xi − ψ(x))
(29)
It was noted in [12] that the Jacobian is given by
dy = |det dTψ (x)| dx =
1 − sψ(x)
n+1
(1 + s(h∇ψ(x), xi − ψ(x)))
det Hess ψ(x) dx.
? ?
It was also noted in [12] that the duality (ψ(s)
)(s) = ψ holds and that therefore,
? (y)
det (dTψ (x)) det dTψ(s)
= 1.
(30)
Now, the next theorem provides a duality formula for an s-concave function ϕ and
ϕ∗(s) . It is a generalization of a duality formula proved for special f in [12].
9
Theorem 7. Let f : (0, ∞) → R be a convex or concave function. Let ϕ : Rn → [0, ∞)
be an s-concave function such that Cϕ is open and bounded, ϕ is differentiable on Cϕ
and that limx→∂ supp(ϕ) ϕs (x) = 0. Then
(s)
(s)
(s)
(s)
Df (Pϕ∗ , Qϕ∗ ) = Df ∗ (Pϕ(s) , Q(s)
ϕ ).
(s)
(31)
(s)
Remark. In particular, if f ≡ 1, or, equivalently, f ∗ = Id, formula (31) becomes
i
h
Z
Z
+ (1 − s) ∇ϕ⊗∇ϕ
det −Hess(ϕ)
2
ϕ
ϕ
.
(32)
(1 + sn) ϕ∗(s) dx =
n+ 1s
h∇ϕ,xi
supp(ϕ)
ϕ 1−s ϕ
Proof. By Definition 4, the change of variable (29) and (30)
Z
det Hess ψ(x)
(s)
∗
Df ∗ Pϕ(s) , Q(s)
=
f
1
1
ϕ
(1 − sψ(x)) s −1 |1 − sψ(x) + shx, ∇ψ(x)i| s +n+1
Rn
1 −1
× 1 − sψ(x) s |1 − s(ψ(x) − hx, ∇ψ(x)i)| dx
Z
det dTψ (x)
=
f∗
1
1
(1 − sψ(x)) s |1 − sψ(x) + shx, ∇ψ(x)i| s
Rn
−1
1 −1
?
1 − sψ(s)
(y)
× 1 − sψ(x) s
dx


Z
?
1− 1
?
 (1 − sψ(s) (y)) s detHess ψ(s) (y) 
=
f 
1 +n+1 
Rn
? (y) + shy, ∇ψ ? (y)i s
−
sψ
1
(s)
(s)
1s −1 ?
?
?
(y) + shy, ∇ψ(s)
(y)i dy
× 1 − sψ(s)
(y)
1 − sψ(s)
(s)
(s)
(s)
= Df Pϕ∗ , Qϕ∗ .
(s)
(s)
The proof of the following entropy inequality for s-concave functions is immediate
with Jensen’s inequality and identity (32).
Theorem 8. Let f : (0, ∞) → R be a convex function. Let ϕ : Rn → (0, ∞) be an
s-concave function such that Cϕ is open and bounded, ϕ is differentiable on Cϕ and that
limx→∂ supp(ϕ) ϕs (x) = 0. Then
(s)
Df
Pϕ(s) , Q(s)
≥ (1 + ns)
ϕ
Z
!
ϕdx f
supp(ϕ)
If f is concave, the inequality is reversed.
10
R
ϕ∗ dx
R (s)
ϕdx
!
.
3
f -divergence for log concave functions.
A function ϕ : Rn → [0, ∞) is log concave, if it is of the form ϕ(x) = e−ψ(x) , where
ψ : Rn → R is a convex function. A log-concave function ϕ can be approximated by the
sequence of k-concave functions {ϕk }∞
k=1
1
ϕk = (1 + k ln ϕ)+k ,
k∈N
(33)
where for a ∈ R, a+ = max{a, 0}. This motivates our definition for f -divergence for log
concave functions. We put
Qϕ = ϕ
and
Pϕ = ϕ−1 e
h∇ϕ,xi
ϕ
det [Hess (− ln ϕ)]
(34)
and define now the f -divergences for log concave functions.
Definition 9. Let f : (0, ∞) → R be a convex or concave function and let ϕ : Rn →
[0, ∞) be a log concave function. Then the f -divergence Df (P ϕ, Qϕ ) of ϕ is
!
h∇ϕ,xi
Z
e ϕ
det [Hess (− ln ϕ)] dx.
(35)
Df (Pϕ , Qϕ ) =
ϕf
ϕ2
supp(ϕ)
Again, we will sometimes write in short Df (ϕ) for Df (Pϕ , Qϕ ).
Remarks and Examples. (i) Similarly to (35),
Z
Df (Qϕ , Pϕ ) =
−1
ϕ
e
h∇ϕ,xi
ϕ
ϕ2
det [−Hess (ln ϕ)] f
supp(ϕ)
e
h∇ϕ,xi
ϕ
!
dx.
det [Hess (− ln ϕ)]
As by (11), Df (Qϕ , Pϕ ) = Df ∗ (Pϕ , Qϕ ), it is enough to consider Df (Pϕ , Qϕ ).
(ii) If we write a log concave function as ϕ = e−ψ , ψ convex, then (35) (and similarly
Df (Qϕ , Pϕ )) can be written as
Z
Df (Pϕ , Qϕ ) =
e−ψ f e2ψ−h∇ψ,xi det [Hessψ] dx.
(36)
supp(ψ)
(iii) Let A be a positive definite, symmetric matrix, C > 0 a constant and ϕ(x) =
Ce−hAx,xi . Then
n
2 det(A)
Cπ n/2
p
Df (Pϕ , Qϕ ) = f
.
(37)
2
C
det(A)
(iv) Let a be a non-zero vector in Rn , C > 0 a constant and let ϕ(x) = Ce−ha,xi . Then
Z
n
C f (0)
Df (Pϕ , Qϕ ) = Qn
e−x dx ,
R
i=1 ai
which is infinity, unless f (0) = 0. Therefore, we require that ϕ = e−ψ is such that ψ is
strictly convex.
If ϕ is an s0 -concave function, then ϕ is s-concave for all s ≤ s0 . In particular, ϕ is
(0)
(0)
(0)
log concave. Thus Df (Pϕ , Qϕ ) is defined for ϕ and Df (Pϕ , Qϕ ) = Df (Pϕ , Qϕ ).
11
On the other hand, as it was remarked in (33), every log concave function can be
approximated by s-concave functions. The next Proposition shows that Definition 9 is
compatible with Definition 4 for s-concave functions.
Proposition 10. Let f : (0, ∞) → R be a convex or concave function. Let s > 0 and
1
for a log concave function ϕ : Rn → [0, ∞) put ϕs = (1 + s ln ϕ)+s . Then
(s)
= Df (Pϕ , Qϕ ).
lim Df Pϕ(s)
, Q(s)
ϕs
s
s→0
1
Proof. Let s > 0 and let ϕs = (1 + s ln ϕ)+s . Then
Z
1
sh∇ϕ, xi
(s)
(s)
(s)
s
Df Pϕs , Qϕs = (1 + s ln ϕ)+ 1 −
ϕ (1 + s ln ϕ)+

h
i
(− ln ϕ)
s∇(ln ϕ)⊗∇(ln ϕ)
s∇ϕ⊗∇ϕ
det Hess
+
−
(1+s ln ϕ)2+
(1+s ln ϕ)2+
ϕ2 (1+s ln ϕ)2+ 

f
 dx.
n+ 1s +1
2
sh∇ϕ,xi
s
(1 + s ln ϕ)+ 1 − ϕ(1+s ln ϕ)
+
Therefore
(s)
lim Df
s→0
(s)
Pϕ(s)
,
Q
= Df (Pϕ , Qϕ ).
ϕ
s
s
Note that we can interchange integration and limit because conditions (6) and (7) hold.
Compare also [6].
Similar to Lemma 6, f -divergences for log concave functions are affine invariant valuations. Also, the proof is similar to the one of Lemma 6 and we omit it.
Corollary 11. Let f : (0, ∞) → R be a convex or concave function and let ϕ : Rn →
[0, ∞) be a log-concave function. Then Df (Pϕ , Qϕ ) is invariant under self adjoint SL(n)
maps and it is a valuation.
Recall that for ϕ : Rn → [0, ∞), the dual function ϕ◦ [5] is defined by
◦
ϕ (x) = infn
y∈R
e−hx,yi
.
ϕ(y)
This definition is connected with the Legendre transform Lϕ(y) = supx∈Rn [hx, yi − ϕ(x)],
namely for ϕ = e−ψ ,
ϕ◦ = e−L(− ln ϕ) = e−L(ψ) .
(38)
Remark. Please observe that Proposition 5 justifies to call Qϕ and Pϕ the cone measures
of the log-concave function ϕ and its polar ϕ◦ .
The next Theorem 3, already mentioned in the introduction, gives a duality relation
for a log concave function and its polar. We will see in Section 5 that it is the functional
analogue of a duality formula for convex bodies.
12
Theorem 3. Let f : (0, ∞) → R be a convex or concave function. Let ϕ : Rn → [0, ∞)
be a log-concave function. Then
Df (Pϕ◦ , Qϕ◦ ) = Df ∗ (Pϕ , Qϕ ).
(39)
Remark. In particular, for f ≡ 1, or, equivalently, f ∗ = Id, formula (39) becomes
Z
Z
h∇ϕ,xi
◦
(40)
ϕ dx =
ϕ−1 (det (Hess (− ln ϕ))) e ϕ .
supp(ϕ)
supp(ϕ)
Proof. We give a direct proof. But please observe that the proof also follows from
Theorem 7, if we let s → 0, together with (38) and Proposition 10.
We write ϕ = e−ψ , ψ convex, and let Lψ(y) be the Legendre transform of ψ. Please
note that when ψ is a C 2 strictly convex function, then
ψ(x) + Lψ(y) = hx, yi if and only if y = ∇ψ(x) if and only if x = ∇Lψ(y).
It follows that
∀y ∈ Rn , ψ(∇Lψ(y)) = hy, ∇Lψ(y)i − Lψ(y)
(41)
∇ψ ◦ ∇Lψ = ∇Lψ ◦ ∇ψ = Id,
(42)
Hess ψ(∇Lψ(y)) Hess Lψ(y) = Id = Hess Lψ(∇ψ(x)) Hess ψ(x).
(43)
and
so that for any x, y ∈ Rn ,
Using equations (41), (42) and (43), the change of variable x = ∇L(ψ(y)) gives
h∇ϕ,xi !
Z
e ϕ
∗
dx
Df ∗ (Pϕ , Qϕ ) =
ϕf
det [Hess (− ln ϕ)]
ϕ2
Rn
−2ψ(x)+h∇ψ,xi Z
e
ψ(x)−h∇ψ,xi
dx
=
det (Hessψ(x)) e
f
det (Hessψ(x))
n
ZR
=
det (Hessψ(∇Lψ(y))) eψ(∇Lψ(y))−hy,∇(Lψ(y))i
Rn
−2ψ(∇Lψ(y))+hy,∇Lψ(y)i e
×f
det (HessLψ(y)) dy
det (Hessψ(∇Lψ(y)))
Z
=
e−Lψ(y) f det(HessLψ(y)) e−hy,∇Lψ(y)i+2Lψ(y) dy
Rn


h∇ϕ◦ ,xi
Z
ϕ◦
e

=
ϕ◦ f det [Hess (− ln ϕ◦ )]
(ϕ◦ )2
Rn
= Df (Pϕ◦ , Qϕ◦ ).
13
A consequence of Theorem 3 is the following entropy inequality for log concave functions. This is Theorem 1 of the introduction.
Theorem 1. Let f : (0, ∞) → R be a convex function and let ϕ : Rn → [0, ∞) be a
log-concave function. Then
!
R ◦ Z
ϕ dx
Df (Pϕ , Qϕ ) ≥ f R
ϕdx .
ϕdx
supp(ϕ)
If f is concave, the inequality is reversed. If f is linear, equality holds. Equality also
holds if ϕ(x) = Ce−hAx,xi , where C is a positive constant and A is a n × n positive
definite matrix.
Proof. The inequality follows immediately from Jensen’s inequality and identity (40).
1
Or, if we approximate ϕ by ϕs = (1 + s ln ϕ)+s , the inequality follows from Theorem 8
letting s → 0.
It is easy to check that equality holds if f is linear and that equality holds for
ϕ(x) = Ce−hAx,xi by (37). In fact, one can assume that A is positive definite and
symmetric.
If we let f (t) = ln t in Theorem 1, we obtain the following corollary which is a
reformulation of Corollary 2 of the introduction.
Corollary 12. Let ϕ : Rn → [0, ∞) be a log-concave function. Then
! R
Z
ϕ◦ dx
D(Pϕ ||Qϕ ) ≤
ϕdx ln R
.
ϕdx
supp(ϕ)
(44)
Equality holds if ϕ(x) = Ce−hAx,xi , where C is a positive constant and A is a n × n
positive definite matrix.
Remarks.
(i) Inequality (44) is stronger than (1). Indeed, as
Z
Z
Z
D(Pϕ ||Qϕ ) = −2 ϕ ln ϕdx − n ϕdx + ϕ ln det (Hess (− ln ϕ)) dx,
inequality (44) is equivalent to
Z
ϕ ln det (Hess (− ln ϕ)) dx
supp(ϕ)
"
! Z
!#
Z
n
◦
≤ 2 Ent(ϕ) + kϕkL1 ln e
ϕdx
ϕ dx
.
supp(ϕ)
supp(ϕ)
Now we apply the functional form of theR Blaschke Santaló inequality [5, 7, 17, 31]. We
assume without loss of generality that ϕdx = 1. Observe that we can also assume
14
R
R
without loss of generality that xϕ(x)dx = 0. If xϕ(x)dx = x0 , replace ϕ by ϕ̃(x) =
ϕ(x + x0 ). We then get,
Z
n
ϕ ln det (Hess (− ln ϕ)) dx ≤ 2 Ent(ϕ) + ln (2πe) ,
supp(ϕ)
which is inequality (1).
(ii) The characterization of equality in (1) now follows by the equality characterization
of the Blaschke Santaló inequality. Indeed, from the arguments in (i), if there is equality
in (1), there is also equality in the functional Blaschke-Santaló inequality. This implies
that the function has the form ϕ(x) = Ce−hAx,xi by [5].
Let us state another corollary to Theorem 1. Its proof follows immediately from Theorem 1 and the functional Blaschke Santaló inequality and its equality characterization.
Corollary 13. Let ϕ : Rn → [0, ∞) be a log-concave function that has center of mass
at 0. Let f : (0, ∞) → R be a convex, decreasing function. Then
! Z
!
(2π)n
ϕdx .
Df (Pϕ , Qϕ ) ≥ f
2
R
supp(ϕ)
ϕdx
If f is a concave, increasing function, the inequality is reversed.
Equality holds in both cases if and only if ϕ(x) = ce−hAx,xi , where c is a positive constant
and A is an n × n positive definite matrix.
4
Applications to special functions.
Now we consider special cases of f -divergences for log concave functions. Please recall that in subsection 2.1, the α-Rényi entropies (15) were introduced as special f divergences. Examples of such Rényi entropies are, for log concave functions ϕ : Rn →
[0, ∞), for f (t) = tλ , −∞ < λ < ∞, the Lλ -affine surface areas asλ (ϕ) of ϕ, introduced
in [12],
!λ
h∇ϕ,xi
Z
e ϕ
det [Hess (− ln ϕ)] dx,
(45)
asλ (ϕ) =
ϕ
ϕ2
supp(ϕ)
or, writing ϕ(x) = e−ψ(x) , ψ convex,
Z
λ
−ψ
asλ (ϕ) = asλ (e ) =
e(2λ−1)ψ(x)−λhx,∇ψ(x)i (det Hess ψ(x)) dx.
(46)
Rn
R
R
Especially, as0 (ϕ) = supp(ϕ) ϕdx and, by (40), as1 (ϕ) = supp(ϕ) ϕ◦ dx. Please note
also that for any log concave function ϕ we have that asλ (ϕ) ≥ 0. Moreover, by Corollary
11, the asλ (ϕ) are affine invariant valuations.
We first want to give a definition for as∞ (ϕ) and as−∞ (ϕ), similarly as it was done
1
for convex bodies [51]. To that end, for λ > 0, let as
˜ λ (ϕ) = (asλ (ϕ)) λ and denote
1
h = (ϕ) λ
e
h∇ϕ,xi
ϕ
ϕ2
det [Hess (− ln ϕ)] .
15
R
Denote by kf kλ =
λ1
the Lλ norm of a function f . Then, for λ → ∞,
! λ1
Z
as
˜ λ (ϕ) =
f λ dx
λ
= khkλ → khk∞ =
h dx.
supp(ϕ)
max
x∈supp(ϕ)
e
h∇ϕ,xi
ϕ
ϕ2
det [Hess (− ln ϕ)].
Therefore, it is natural to put
as∞ (ϕ) =
max
x∈supp(ϕ)
e
h∇ϕ,xi
ϕ
det [Hess (− ln ϕ)] .
ϕ2
(47)
Similarly, for λ → −∞,
1
as
˜ λ (ϕ) →
maxx∈supp(ϕ)
=
h∇ϕ,xi
e ϕ
ϕ2
det [Hess (− ln ϕ)]
ϕ2
min
= as−∞ (ϕ).
h∇ϕ,xi
x∈supp(ϕ)
e ϕ det [Hess (− ln ϕ)]
Hence we have that
as−∞ (ϕ) =
(48)
1
.
as∞ (ϕ)
It is also easy to see that these expressions are invariant under symmetric affine transformations with determinant 1.
The next theorem gives the analogue, for log concave functions, of a monotonicity
behavior of the Lλ -affine surface area that was established for convex bodies in [41, 69].
The case β = 0 and α = 1 was already proved in [12].
Proposition 14. Let α 6= β, λ 6= β be real numbers. Let ϕ : Rn → [0, ∞) be a log
concave function.
λ−β
α−λ
α−β
(i) If 1 ≤ α−β
asβ (ϕ) α−β .
λ−β < ∞, then asλ (ϕ) ≤ asα (ϕ)
R α−λ
λ
ϕ α .
< ∞, then asλ (ϕ) ≤ (asα (ϕ)) α
λ−β
(iii) If β ≤ λ, then asλ (ϕ) ≤ as∞ (ϕ)
asβ (ϕ).
(ii) If 1 ≤
α
λ
α
If α−β
λ−β = 1 in (i), respectively λ = 1 in (ii), then α = λ and equality holds trivially in
(i) respectively (ii). Equality also holds if ϕ(x) = Ce−hAx,xi .
Proof. The proofs follow by Hölder’s inequality, which, in (i), enforces the condition
α−β
λ−β > 1. The case(ii) is a special case of (i) for β = 0. We show (i). The others follow
16
similarly.
Z
asλ (ϕ)
=
e
ϕ
supp(ϕ)
Z "
ϕ
=
e
!λ
h∇ϕ,xi
ϕ
det [Hess (− ln ϕ)]
ϕ2
dx
λ−β
!α # α−β
h∇ϕ,xi
ϕ
det [Hess (− ln ϕ)]
ϕ2

e
× ϕ
α−λ
!β  α−β
h∇ϕ,xi
ϕ
det [Hess (− ln ϕ)]
ϕ2
dx

λ−β
α−λ
asα (ϕ) α−β asβ (ϕ) α−β .
≤
It follows from Proposition 14 (ii) that for 0 < λ ≤ α,
0≤
asλ (ϕ)
R
ϕdx
λ1
≤
which means that for λ > 0 the function λ →
is increasing for λ > 0. Therefore, the limit
Ωϕ = lim
λ↓0
asα (ϕ)
R
ϕdx
α1
asRλ (ϕ)
dx
ϕ
asλ (ϕ)
R
ϕdx
,
λ1
is bounded below by 0 and
λ1
(49)
exists and the quantity Ωϕ is an affine invariant. It is the analogue for log concave
functions of an affine invariant introduced by Paouris and Werner in [54] for convex
bodies. The quantity Ωϕ is related to the relative entropy as follows.
Proposition 15. Let ϕ : Rn → [0, ∞) be a log concave function. Then
Ωϕ = lim
λ↓0
asλ (ϕ)
R
ϕdx
λ1
= lim
λ↑0
asλ (ϕ)
R
ϕdx
17
λ1
= exp
D(Pϕ ||Qϕ )
R
ϕdx
.
Proof. By definition and de l’Hôspital,
1
asλ (ϕ) λ
1
asλ (ϕ)
R
= lim
= lim exp
ln R
λ↓0
λ↓0
λ
ϕ
ϕdx

λ 
h∇ϕ,xi
R d
e ϕ
det [Hess (− ln ϕ)]
dx 

dλ ϕ
ϕ2


= exp lim

asλ (ϕ)
 λ↓0

Ωϕ
R
=
 ϕ ln
exp 

=
exp
e
h∇ϕ,xi
ϕ
ϕ2
D(Pϕ ||Qϕ )
R
ϕdx

det [Hess (− ln ϕ)] dx 

R

ϕdx
.
It also follows from Proposition 14 (ii) that for λ < 0, the function λ →
λ1
increasing We compute limλ↑0 asRλϕ(ϕ)
as above.
◦
λ1
asRλ (ϕ)
dx
ϕ
is
Corollary 16. Let ϕ : Rn → [0, ∞) be a log concave function.
λ1
λ1
as
R λ (ϕ)
R λ (ϕ)
for
all
λ
>
0
and
Ω
≥
for all λ < 0.
(i) Ωϕ ≤ as
ϕ
ϕdx
ϕdx
(ii) Ωϕ Ωϕ◦ ≤ 1.
(iii) Ωϕ = limα→1
◦
as
Rα (ϕ )
ϕdx
1
1−α
.
Equality holds in (i) and (ii) if ϕ = Ce−hAx,xi .
Proof. (i) is deduced immediately from the monotonicity behavior of the function λ →
λ1
asRλ (ϕ)
dx
and the definition of Ωϕ .
ϕ
R
ϕ◦ dx
◦
R 1 (ϕ) = R
R 1 (ϕ ) =
(ii) By (i), Ωϕ ≤ as
and Ωϕ◦ ≤ as
ϕdx
ϕdx
ϕ◦ dx
the bipolar property (ϕ◦ )◦ = ϕ. Thus (ii) follows.
R
R ϕdx .
ϕ◦ dx
Here, we have also used
(iii) We use the duality formula asλ (ϕ) = as1−λ (ϕ◦ ) which was first proved in [12]. Note
that it can also be obtained as a special case of Theorem 3 for f (t) = tλ . By definition
Ωϕ◦
=
lim
λ→0
asλ (ϕ)◦
R
ϕ◦
Therefore, Ωϕ = limα→1
λ1
◦
as
Rα (ϕ )
ϕdx
= lim
λ→0
1
1−α
as1−λ (ϕ)
R
ϕ◦
.
18
λ1
= lim
α→1
asα (ϕ)
R
ϕ◦
1
1−α
.
5
Application to convex bodies.
Let us now consider the case of 2-homogeneous functions ψ, that is ψ(λx) = λ2 ψ(x)
for any λ ∈ R+ and x ∈ Rn . Such functions ψ are necessarily (and this is obviously
sufficient) of the form ψ(x) = kxk2K /2 for a certain convex body K with 0 in its interior,
where we have denoted by k.kK the gauge function of K,
kxkK = min{λ ≥ 0 : λx ∈ K} = max◦ hx, yi = hK ◦ (x).
y∈K
Differentiating with respect to λ at λ = 1, we get
hx, ∇ψ(x)i = 2ψ(x).
(50)
Now we apply this function to the identities and inequalities which we have obtained
for f -divergences for log concave functions. It was already observed in [12] that the
Lλ -affine surface area for log concave functions is a generalization of Lλ -affine surface
area for convex bodies,
Indeed,
it was noted there that if one applies the log concave
k·k2K
to Definition 45, then one obtains Lλ -affine surface area for
function ϕK = exp − 2
convex bodies. Please note also that
Z
Z
n
n
kxk2 ◦
kxk2
(2π) 2 |K|
(2π) 2 |K ◦ |
K
− 2K
−
2
e
dx =
and
e
dx =
.
(51)
|B2n |
|B2n |
Recall that B2n denotes the n-dimensional Euclidean unit ball, and for a convex body K
in Rn , K ◦ is the polar of K (20) and |K| is its volume.
The following is a generalization of a result in [12] but it is proved in exactly the
same way. We include the proof for completeness.
k·k2
Theorem 17. Let K be a convex body in Rn with 0 in its interior. Let ϕK = exp − 2K
and f : (0, ∞) → R be a convex or concave function. Then
n
Df (PϕK , QϕK ) =
(2π) 2
Df (PK , QK ).
n|B2n |
Here, PK and QK are as in (19) and, for ϕK , PϕK and QϕK are as in (34).
k·k2
Proof. We will use formula (36) for ψ = 2K and integrate in polar coordinates with
PK
respect to the normalized cone measure n|K|
(19) of K. Thus, if we write x = rz,
with z ∈ ∂K, dx = rn−1 drdPK (z). We also use that the map x 7→ det Hess ψ(x) is
0-homogeneous. Therefore, with (50),
Z
Z +∞
−r 2
Df (PϕK , QϕK ) =
rn−1 e 2 dr
f (det Hess ψ(z)) dPK (z)
0
∂K
n Z
2π) 2
f (det Hess ψ(z)) dPK (z).
=
n|B2n | ∂K
It was proved in [12] that for all z ∈ ∂K,
det (Hessz ψ) =
κK (z)
.
hz, NK (z)in+1
19
(52)
Observe that for the GK (z) introduced in this lemma, kGK (z)kK ◦ = hz, NK (z)i. Thus
n Z
(2π) 2
κ(x)
hx, NK (x)idµK (x)
Df (PϕK , QϕK ) =
f
n|B2n | ∂K
hx, NK (x)in+1
n
=
(2π) 2
Df (PK , QK ).
n|B2n |
k·k2
Now we apply Theorem 1 to ϕ = exp − 2K and we obtain the following inequalities. Those were already proved, with different methods, in [68]. In fact, it was shown
there that equality holds if and only if K is an ellipsoid.
Corollary 18. Let K be a convex body in Rn with the origin in its interior. Let f :
(0, ∞) → R be a convex function. Then
◦ |K |
.
(53)
Df (PK , QK ) ≥ n |K|f
|K|
If f is concave, the inequality is reversed. Equality holds if K is an ellipsoid.
k·k2
Proof. Let ϕ = exp − 2K and let f be convex. By Theorem 1, together with (51),
Z
supp(ϕ)
e−
kxk2
K
2
f
◦ n
kxk2K
(2π) 2 |K|
|K |
det Hess
dx ≥
f
.
2
|B2n |
|K|
Now we use again (52) and, as above, make the change of variable, x = rz, z ∈ ∂K.
Then this becomes
◦ Z
κ(x)
|K |
f
.
hx,
N
(x)idµ
(x)
≥
n
|K|f
K
K
n+1
hx, NK (x)i
|K|
∂K
For f concave, the direction in the inequality changes.
k·k2
In this way, by applying them to the particular log concave function exp − 2K ,
many of the known inequalities for convex bodies can be deduced from the corresponding
ones for log concave functions. Examples include:
p
(i) For −∞ ≤ p ≤ ∞, p 6= −n, the function f (t) = t n+p is concave forp ≥ 0 and
convex
k·k2K
for p ≤ 0. Then, as a consequence of Theorem 1 applied to ϕ = exp − 2 , and the
Blaschke Santaló inequality [10, 49, 59] we obtain the
isoperimetric inequalities
Lp -affine
of [41, 69]. If we apply Proposition 14 to ϕ = exp −
k·k2K
2
, we obtain the monotonicity
behavior of the Lp -affine surface area for convex bodies proved in [41, 69].
20
k·k2
(ii) If we apply Proposition 15 and Corollary 16 to ϕ = exp − 2K , then we obtain
entropy inequalities first proved in [54] for convex bodies. E.g, Corollary 16, together
with the Blaschke Santaló inequality gives the following isoperimetric inequality of [54]
Ω K ◦ ≤ Ω
n
B2
◦ = |B n |2n .
2
1
n| n
|B2
k·k2
Finally, the duality formula (39) applied to ϕ = exp − 2K corresponds to a duality formula for convex bodies and is a generalization of previously established duality
formulas [27, 69] for convex bodies, asp (K) = as n2 (K ◦ ). See also [37]. We skip the
p
proof.
Proposition 19. Let K be a convex body in Rn and let f : (0, ∞) → R be a convex or
concave function. Then
Df (PK ◦ , QK ◦ ) = Df ∗ (PK , QK ).
6
Linearization.
In this section we linearize the inequalities of Corollary 13 around its equality case.
We treat only one inequality. The other one is done in the same way. We rewrite the
inequality in terms of a convex function ψ : Rn → R such that ϕ = e−ψ and get
Z
Z
(2π)n
−ψ
−ψ
2ψ−h∇ψ,xi
R
e dx .
(54)
e f e
det(Hess(ψ)) dx ≥ f
( Rn e−ψ dx)2
Rn
Rn
Corollary 13 requires that ϕ has center of mass at the origin. This is the case if ψ
is even. We then linearize around the equality case ψ(x) = kxk2 /2 and obtain the
following functional inequalities. See also [6], [25], [26]. The proof, which we include
for completeness, follows [6]. Throughout, k.kHS denotes the Hilbert Schmidt norm and
4η = tr(Hess η) is the Laplacian of η. γn is the normalized Gaussian measure on Rn
2
R
R
and Varγn (η) = Rn η 2 dγn − Rn ηdγn is the variance of η.
Corollary 20. Let η ∈ C 2 (Rn ) ∩ L2 (Rn , γn ) be even. Then
R
R
2
1
(i)
kHess ηk2HS dγn .
2 Rn (4η − h∇η, xi) dγn ≤
Rn
Z
1
2
(4η − h∇η, xi) dγn ≤ Varγn (η) ≤
k∇ηk dγn −
4
n
n
R
R
Z
Z
Z
1
1
2
2
k∇ηk dγn −
(4η − h∇η, xi) dγn +
kHessηk2HS dγn .
2 Rn
2 Rn
Rn
Z
(ii)
2
Remark. The left hand side of Corollary 20 (ii) together with Corollary 20 (i) gives the
following reverse Poincaré inequality obtained in [6] (see also [25], [26])
Z kHess ηk2HS
2
k∇ηk −
dγn ≤ Varγn (η).
2
Rn
21
Proof. We first prove the corollary for functions with bounded support. Thus, let η
be an even, twice continuously differentiable function with bounded support and let
ψ(x) = kxk2 /2 + εη(x). Note that for sufficiently small ε the function ψ is convex and
that ψ is even as η is even.Therefore we can plug ψ into inequality (54) and develop in
powers of ε. We evaluate first the right hand expression of (54).
! Z
(2π)n
−kxk2 /2−εη
e
f
dx
R
2
Rn
e−kxk2 /2−εη dx
Rn
Z
2 !
Z
Z
= (2π)n/2 f 1 + 2ε
ηdγn − ε2
η 2 dγn + 3ε2
ηdγn
Rn
Rn
Rn
Z
1−ε
ηdγn +
Rn
ε2
2
Z
η 2 dγn
+ O(ε3 ).
Rn
00
As f (1 + t) = f (1) + f 0 (1)t + f 2(1) t2 + O(t3 ), we get that the right hand side of (54)
equals
Z
Z
f (1)
0
n/2
0
2
2
− f (1) +
(2π)
f (1) + ε [2f (1) − f (1)]
ηdγn + ε
η dγn
2
Rn
Rn
Z
2
ηdγn [f 0 (1) + 2f 00 (1)]
+ O(ε3 ).
Rn
We evaluate now the left hand expression of (54). Since Hess ψ = I + ε Hess η, we
obtain for the left hand side
Z
2
e−kxk /2−εη f eε(2η−h∇η,xi) det (I + ε Hess η) dx.
Rn
By Taylor’s theorem this equals
Z
2
ε2
ε2
1 + ε(2η − h∇η, xi) + (2η − h∇η, xi)2
e−kxk /2 1 − εη + η 2 · f
2
2
Rn
·det(I + ε Hess η) dx + O(ε3 ).
For a matrix A = (ai,j )i,j=1,...,n , let D(A) =
Pn
i=1
Pn
j6=i [ai,i aj,j
− a2i,j ]. Note that each
2
2 × 2 minor is counted twice. Then det(I + ε Hess η) = 1 + ε4η + ε2 D(Hess η) + O(ε3 ).
22
Therefore the left hand side of (54) equals
Z ε2
(2π)n/2
1 − εη + η 2 f 1 + ε (2η + 4η − h∇η, xi)
2
Rn
!
2
(2η − h∇η, xi)
D(Hess η)
2
+ε
+ 4η (2η − h∇η, xi) +
dγn + O(ε3 ) =
2
2
Z
Z
n/2
0
(2π)
f (1) + ε −f (1)
ηdγn + f (1)
(2η − h∇η, xi + 4η)dγn
Rn
Rn
Z
Z
η2
0
2
dγn − f (1)
η (2η − h∇η, xi + 4η) dγn +
+ε f (1)
Rn
Rn 2
!
!
Z
2
(2η
−
h∇η,
xi)
D(Hess
η)
f 0 (1)
+ 4η (2η − h∇η, xi) +
dγn
2
2
Rn
Z
f 00 (1)
2
+
(2η − h∇η, xi + 4η) dγn
+ O(ε3 ).
2
n
R
R
Now observe that Rn (4η − h∇η, xi)dγn = 0. Also, as the coefficients of order zero and
of order ε are the same on the left hand side and the right hand side, we discard them.
We divide both sides by ε2 and take the limit for ε → 0. Thus, the inequality (54) is
equivalent to
#
"Z
Z
2
f 0 (1) + 2f 00 (1)
ηdγn
Rn
η 2 dγn ≤
dγn −
Rn
Z
Z
Z
f 0 (1)
D(Hess η) dγn
f (1) + 2f (1)
η4η dγn −
ηh∇η, xi dγn +
2
Rn
Rn
Rn
Z
Z
Z
h∇η, xi2
f 00 (1)
2
0
00
(4η) dγn + f (1) + f (1)
dγn −
4ηh∇η, xidγn .
+
2
2
Rn
Rn
Rn
R
R
Integration by parts yields Rn ηh∇η, xidγn = 12 Rn η 2 (x) (kxk2 − n) dγn and
Z
Z
Z
1
η4η dγn = −
k∇ηk2 dγn +
η 2 (x) (kxk2 − n) dγn .
2 Rn
Rn
Rn
0
00
We put a = f 0 (1) and b = f 00 (1). Note that a ≤ 0 and b ≥ 0, as f is convex and
decreasing. Thus, the inequality becomes
Z
2
a + 2b Varγn (η) −
k∇ηk dγn
Rn
Z
Z
a
a+b
2
2
≥
kHess ηkHS dγn −
(4η − h∇η, xi) dγn .
2 Rn
2
n
R
Hence we have shown that the inequality holds for all twice continuously differentiable functions η with bounded support. Now we extend it to all twice continuously
differentiable functions η ∈ L2 (Rn , γn ) satisfying the necessary integrability conditions,
by a standard approximation argument, as follows.
Let χk be a twice continuously differentiable function bounded between zero and one
such that χn (x) = 1 for all kxk ≤ k and χn (x) = 0 for all kxk > k + 1. Then, for all
23
k∈N
Z
a + 2b Varγn (η · χk ) −
2
k∇(η · χk )k dγn
Rn
≥
a
2
Z
Rn
kHess (η · χk )k2HS dγn −
which is equivalent to
"Z
2
(η · χk )dγn
2b
Z
2
dγn ,
4(η · χk ) − h∇(η · χk ), xi
Rn
#
Z
k∇(η · χk )k2 dγn
+
Rn
a+b
2
Rn
#
2
1
dγn
4(η · χk ) − h∇(η · χk ), xi
(η · χk ) dγn +
− a
Rn
Rn 2
"Z
#
2 Z
Z
2
kHess
ηk
HS
(η · χk )dγn +
k∇(η · χk )k2 dγn +
≤ (−a)
dγn
2
Rn
Rn
Rn
"Z
#
2
Z
1
2
+ b
2(η · χk ) dγn +
4(η · χk ) − h∇(η · χk ), xi
dγn .
Rn
Rn 2
"Z
Z
2
Now we pass to the limit k → ∞ on both sides and obtain
"
#
Z
2
Z
2
2b lim inf
(η · χk )dγn + lim inf
k∇(η · χk )k dγn
k→∞
k→∞
Rn
Rn
#
2
1
4(η · χk ) − h∇(η · χk ), xi
dγn
− a lim inf
(η · χk ) dγn + lim inf
k→∞
k→∞
Rn
Rn 2
"
#
Z
2
Z
2
≤ (−a) lim sup
(η · χk )dγn + lim sup
k∇(η · χk )k dγn +
"
Z
2
k→∞
"
b
Z
Rn
Z
− a lim sup
k→∞
k→∞
Rn
2(η · χk )2 dγn + lim sup
lim sup
k→∞
Z
Rn
Z
k→∞
Rn
Rn
1
4(η · χk ) − h∇(η · χk ), xi
2
#
2
dγn
kHess ηk2HS
dγn .
2
Fatou’s lemma and the dominated convergence theorem yield
Z
Z
a
a+b
2
kHess ηk2HS dγn −
(4η − h∇η, xi) dγn
2 Rn
2
Rn
Z
2
≤ a + 2b Varγn (η) −
k∇ηk dγn .
Rn
Finally, we consider the cases a + 2b = 0, a + 2b > 0 and a + 2b < 0 and optimize in each
case.
24
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Umut Caglar
Department of Mathematics
Case Western Reserve University
Cleveland, Ohio 44106, U. S. A.
[email protected]
Elisabeth Werner
Department of Mathematics
Case Western Reserve University
Cleveland, Ohio 44106, U. S. A.
[email protected]
Université de Lille 1
UFR de Mathématique
59655 Villeneuve d’Ascq, France
29