orlicz centroid bodies erwin lutwak, deane yang and

ORLICZ CENTROID BODIES

Erwin Lutwak, Deane Yang and Gaoyong Zhang

Abstract

The sharp affine isoperimetric inequality that bounds the vol-ume of the centroid body of a star body (from below) by thevolume of the star body itself is the Busemann-Petty centroid in-equality. A decade ago, the Lp analogue of the classical Busemann-Petty centroid inequality was proved. Here, the definition of thecentroid body is extended to an Orlicz centroid body of a starbody, and the corresponding analogue of the Busemann-Petty cen-troid inequality is established for convex bodies.

The centroids of the intersections of an origin-symmetric body withhalf-spaces form the surface of a convex body. This “centroid body” isa concept that dates back at least to Dupin.

The classical affine isoperimetric inequality that relates the volumeof a convex body with that of its centroid body was conjectured byBlaschke and established in a landmark work by Petty [53]. Petty’s in-equality became known as the Busemann-Petty centroid inequality be-cause, in establishing his inequality, Petty not only made critical use ofBusemann’s random simplex inequality, but as Petty stated, he “rein-terpreted” it. (See, e.g., the books by Gardner [12], Leichtweiss [25],Schneider [55], and Thompson [58] for reference.)

The concept of a centroid body had a natural extension in whatbecame known as the Lp Brunn-Minkowski theory and its dual. Thistheory had its origins in the early 1960s when Firey (see, e.g., Schneider[55]) introduced his concept of Lp compositions of convex bodies. Threedecades later, in [34] and [35] these Firey-Minkowski Lp combinationswere shown to lead to an embryonic Lp Brunn-Minkowski theory. Thistheory (and its dual) has witnessed a rapid growth. (See, e.g., [1–9,17–23,26–32,34–44,46–48,54,56,57,59,62].)

The Lp analogues of centroid bodies became a central focus withinthe Lp Brunn-Minkowski theory and its dual and establishing the Lp-analogue of the Busemann-Petty centroid inequality became a majorgoal. This was accomplished by the authors of the present paper in [37]with an independent approach presented by Campi and Gronchi [3].The Lp centroid bodies quickly became objects of interest in asymptotic

Mathematics Subject Classification. MR 52A40.Research supported, in part, by NSF Grant DMS–0706859.

PROOF COPY 1 NOT FOR DISTRIBUTION

2 ERWIN LUTWAK, DEANE YANG AND GAOYONG ZHANG

geometric analysis (see, e.g., [10], [11], [24], [49], [50], [51], [52]) and eventhe theory of stable distributions (see, e.g., [48]).

Using concepts introduced by Ludwig [29], Haberl and Schuster [21]were led to establish “asymmetric” versions of the Lp Busemann-Pettycentroid inequality that, for bodies that are not origin-symmetric, arestronger than the Lp Busemann-Petty centroid inequality presented in[37] and [3]. The “asymmetric” inequalities obtained by Haberl andSchuster are ideally suited for non-symmetric bodies. This can be seenby looking at the Haberl-Schuster version of the Lp analogue of theclassical Blaschke-Santalo inequality that was presented in [46]. Whilefor origin symmetric bodies, the Lp extension of [46] does recover theoriginal Blaschke-Santalo inequality as p→∞, for arbitrary bodies onlythe Haberl-Schuster version does so.

The works of Haberl and Schuster [21] (see also [22]), Ludwig andReitzner [32], and Ludwig [31] have demonstrated the clear need tomove beyond the Lp Brunn-Minkowski theory to what we are callingan Orlicz Brunn-Minkowski theory. This need is not only motivated bycompelling geometric considerations (such as those presented in Ludwigand Reitzner [32]), but also by the desire to obtain Sobolev bounds(see [22]) of a far more general nature.

This paper is the second in a series intended to develop a few of theelements of an Orlicz Brunn-Minkowski theory and its dual. Here wedefine the Orlicz centroid body, establish some of its basic properties,and most importantly establish (what we call) the Orlicz Busemann-Petty centroid inequality (for Orlicz centroid bodies).

We consider convex φ : R → [0,∞) such that φ(0) = 0. This meansthat φ must be decreasing on (−∞, 0] and increasing on [0,∞). Werequire that one of these is happening strictly so; i.e., φ is either strictlydecreasing on (−∞, 0] or strictly increasing on [0,∞). The class of suchφ will be denoted by C.

If K is a star body (see Section 1 for precise definitions) with respectto the origin in Rn with volume |K|, and φ ∈ C then we define the Orliczcentroid body ΓφK of K as the convex body whose support function atx ∈ Rn is given by

h(ΓφK;x) = inf{λ > 0 :

1|K|

∫Kφ(x · y

λ

)dy ≤ 1

},

where x ·y denotes the standard inner product of x and y in Rn and theintegration is with respect to Lebesgue measure in Rn.

When φp(t) = |t|p, with p ≥ 1, then

ΓφpK = ΓpK,

PROOF COPY NOT FOR DISTRIBUTION

ORLICZ CENTROID BODIES 3

where ΓpK is the Lp centroid body of K, whose support function isgiven by

h(ΓpK;x)p =1|K|

∫K|x · y|p dy.

For p = 1 the body ΓpK is the classical centroid body, ΓK, of K.We will establish the following affine isoperimetric inequality for Or-

licz centroid bodies.

Theorem. If φ ∈ C and K is a convex body in Rn that contains theorigin in its interior, then the volume ratio

|ΓφK|/|K|

is minimized if and only if K is an ellipsoid centered at the origin.

The theorem contains as a special case the classical Busemann-Pettycentroid inequality for convex bodies [53], as well as the Lp Busemann-Petty centroid inequality for convex bodies that was established in [37]and [3] and the “asymmetric” version of the Lp Busemann-Petty cen-troid inequality for convex bodies that was established by Haberl andSchuster [21].

For quick later reference, we list in Section 1 some basic, and forthe most part well-known, facts regarding convex bodies. The basicproperties of the Orlicz centroid operator are developed in Section 2.In Section 3 the Theorem is established. Section 4 concludes with someopen problems.

1. Basics regarding convex and star bodies

The setting will be Euclidean n-space Rn. We write e1, . . . , en for thestandard orthonormal basis of Rn and when we write Rn = Rn−1 × Rwe always assume that en is associated with the last factor.

We will attempt to use x, y for vectors in Rn and x′, y′ for vectors inRn−1. We will also attempt to use a, b, s, t for numbers in R and c, λ forstrictly positive reals. If Q is a Borel subset of Rn and Q is contained inan i-dimensional affine subspace of Rn but in no affine subspace of lowerdimension, then |Q| will denote the i-dimensional Lebesgue measure ofQ. If x ∈ Rn then by abuse of notation we will write |x| for the normof x.

For A ∈ GL(n) write At for the transpose of A and A−t for the inverseof the transpose (contragradient) of A. Write |A| for the absolute valueof the determinant of A.

We say that a sequence {φi}, of φi ∈ C, is such that φi → φo ∈ Cprovided

|φi − φo|I := maxt∈I|φi(t)− φo(t)| −→ 0,



for every compact interval I ⊂ R. For φ ∈ C define φ? ∈ C by

(1.1) φ?(t) =∫ 1

0φ(ts) dsn,

where dsn = nsn−1ds. Obviously, φi → φo ∈ C implies φ?i → φ?o.Associated with each φ ∈ C is cφ ∈ (0,∞) defined by

cφ = min{c > 0 : max{φ(c), φ(−c)} ≤ 1}.

Let ρ(K; ·) = ρK : Rn \ {0} → [0,∞) denote the radial function ofthe set K ⊂ Rn, star-shaped about the origin; i.e. ρK(x) = max{λ >0 : λx ∈ K}. If ρK is strictly positive and continuous, then we call K astar body and we denote the class of star bodies in Rn by Sno . If c > 0,then obviously for the dilate cK = {cx : x ∈ K} we have

(1.2) ρcK = cρK .

The radial distance between K,L ∈ Sno is

|ρK − ρL|∞ = maxu∈Sn−1

|ρK(u)− ρL(u)|.

Let h(K; ·) = hK : Rn → R denote the support function of the convexbody (compact convex subset) K in Rn; i.e., h(K;x) = max{x · y : y ∈K}. The Hausdorff distance between the convex bodies K and L is

|hK − hL|∞ = maxu∈Sn−1

|hK(u)− hL(u)|.

We write Kn for the space of convex bodies of Rn. We write Kno for theset of convex bodies that contain the origin in their interiors. On Knothe Haudorff metric and the radial metric induce the same topology.

We shall require the obvious facts that for K,L ∈ Kn, we have

(1.3) K ⊂ L if and only if hK ≤ hL,

and that for c > 0 and x ∈ Rn,

(1.4) hcK(x) = chK(x) and hK(cx) = chK(x).

More generally, from the definition of the support function it followsimmediately that for A ∈ GL(n) the support function of the imageAK = {Ay : y ∈ K} of K is given by

(1.5) hAK(x) = hK(Atx).

For K ∈ Sno , define the real numbers RK and rK by

(1.6) RK = maxu∈Sn−1

ρK(u) and rK = minu∈Sn−1

ρK(u).

Note that the definition of Sno is such that 0 < rK ≤ RK < ∞, for allK ∈ Sno .



Throughout, B = {x ∈ Rn : |x| ≤ 1} will denote the unit ball centeredat the origin, and ωn = |B| will denote its n-dimensional volume. Weshall make use of the trivial fact that for uo ∈ Sn−1,

(1.7) ωn−1 =∫Sn−1

(uo · u)+ dS(u) =12

∫Sn−1

|uo · u| dS(u),

where (t)+ = max{t, 0} for t ∈ R, and where S denotes Lebesgue mea-sure on Sn−1; i.e., S is (n− 1)-dimensional Hausdorff measure.

For a convex body K and a direction u ∈ Sn−1, let Ku denote theimage of the orthogonal projection of K onto u⊥, the subspace of Rn

orthogonal to u. We write ` u(K; ·) : Ku → R and ` u(K; ·) : Ku → Rfor the undergraph and overgraph functions of K in the direction u; i.e.

K ={y′ + tu : −` u(K; y′) ≤ t ≤ ` u(K; y′) for y′ ∈ Ku

}.

Thus the Steiner symmetral SuK of K ∈ Kno in direction u can bedefined as the body whose orthogonal projection onto u⊥ is identical tothat of K and whose undergraph and overgraph functions are given by

` u(SuK; y′) =12

[` u(K; y′) + ` u(K; y′)](1.8a)

and

` u(SuK; y′) =12

[` u(K; y′) + ` u(K; y′)].(1.8b)

For y′ ∈ Ku, define my′ = my′(u) by

my′(u) =12

[` u(K; y′)− ` u(K; y′)]

so that the midpoint of the chord K ∩ (y′ + Ru) is y′ + my′(u)u. Thelength |K ∩ (y′ + Ru)| of this chord will be denoted by σy′ = σy′(u).Note that the midpoints of the chords of K in the direction u lie in asubspace if and only if there exists an x′o ∈ Ku such that

x′o · y′ = my′ , for all y′ ∈ Ku.

In this case {y′− ` u(K; y′)u : y′ ∈ relintKu}, the undergraph of K withrespect to u, is mapped into the overgraph by the linear transformation

y′ + tu 7−→ y′ + [2(x′o · y′)− t]u.A classical characterization of the ellipsoid is the following: A convex

body K ∈ Kno is an origin centered ellipsoid if and only if for eachdirection u ∈ Sn−1 all of the midpoints of the chords of K parallel to ulie in a subspace of Rn. Gruber [15] showed how the following Lemmais a consequence of the Gruber-Ludwig theorem [16]

Lemma 1.1. A convex body K ∈ Kno is an origin centered ellipsoid ifand only if there exists an εK > 0 such that for each direction u ∈ Sn−1

all of the chords of K that come within a distance of εK of the originand are parallel to u, have midpoints that lie in a subspace of Rn.



When considering the convex body K ⊂ Rn−1×R, for (x′, t) ∈ Rn−1×R we will usually write h(K;x′, t) rather than h(K; (x′, t)).

The following is well known:

Lemma 1.2. Suppose K ∈ Kno and u ∈ Sn−1. For y′ ∈ relintKu, theovergraph and undergraph functions of K in direction u are given by

` u(K; y′) = minx′∈u⊥

{h(K;x′, 1)− x′ · y′

},(1.9a)

and

` u(K; y′) = minx′∈u⊥

{h(K;x′,−1)− x′ · y′

}.(1.9b)

See [3] for an application of (1.9a) and (1.9b) to the proof of the LpBusemann-Petty centroid inequality.

Proof. Suppose x′ ∈ u⊥. From the definition of the overgraph itfollows immediately that y′ + ` u(K; y′)u ∈ K, and thus the definitionof the support function shows that

(y′ + ` u(K; y′)u) · (x′ + u) ≤ hK(x′ + u).

Thus,x′ · y′ + ` u(K; y′) ≤ hK(x′ + u) = h(K;x′, 1),

for all x′ ∈ u⊥.Since K has a support hyperplane at y′+ ` u(K; y′)u ∈ ∂K, and since

y′ ∈ relintKu, there exists a vector of the form x′o +u, with x′o ∈ u⊥, sothat

(y′ + ` u(K; y′)u) · (x′o + u) = hK(x′o + u) = h(K;x′o, 1).

Therefore,` u(K; y′) = min

x′∈u⊥

{h(K;x′, 1)− x′ · y′

}.

Formula (1.9b) can be shown in the same way. q.e.d.

We shall require the following crude estimate.

Lemma 1.3. Suppose K ∈ Kno and u ∈ Sn−1. If y′ ∈ (rK/2)B ∩ u⊥,and x′1, x

′2 ∈ u⊥ are such that

` u(K; y′) = h(K;x′1, 1)− x′1 · y′

and` u(K; y′) = h(K;x′2,−1)− x′2 · y′,

then both

|x′1|, |x′2| ≤2RKrK

.



Proof. Note that since y′ ∈ (rK/2)B∩u⊥, it follows that both ` u(K; y′) >0 and ` u(K; y′) > 0.

Observe that since rKB ⊂ K, we have from (1.3),

(1.10) hK

((x′1, 1)√1 + |x′1|2

)≥ rK .

From the fact that K contains the origin in its interior, the definitionof RK and definition (1.9a), the hypothesis, (1.10) and (1.4), we have

RK ≥ ` u(K; y′)

= h(K;x′1, 1)− x′1 · y′

≥ rK(1 + |x′1|2)1/2 − x′1 · y′

≥ rK |x′1| − x′1 · y′

≥ rK |x′1| −rK2|x′1|,

where the last step comes from the hypothesis that |y′| ≤ rK/2.The estimate for |x′2| can be established in the identical manner.

q.e.d.

2. Definition and basic properties of Orlicz centroid bodies

If φ ∈ C, then the Orlicz centroid body ΓφK of K ∈ Sno is defined asthe body whose support function is given by

(2.1) hΓφK(x) = inf{λ > 0 :

1|K|

∫Kφ(x · y

λ

)dy ≤ 1

},

where the integration is with respect to Lebesgue measure on Rn. Ob-serve that since lims→∞ φ(s) = ∞ or lims→−∞ φ(s) = ∞ we havehΓφK(x) > 0 whenever x 6= 0.

It will be helpful to also use the alternate definition:

(2.2) hΓφK(x) = inf{λ > 0 :

∫Sn−1

φ?( 1λ(x · u)ρK(u)) dV ∗K(u) ≤ 1

},

where φ? is defined by (1.1) and dV ∗K is the volume-normalized dualconical measure of K, defined by

|K|dV ∗K =1nρnK dS,

where S is Lebesgue measure on Sn−1 (i.e., (n − 1)-dimensional Haus-dorff measure). We shall make use of the fact that the volume-normalizeddual conical measure

(2.3) V ∗K is a probability measure on Sn−1.



The equivalence of the two definitions is a consequence of the fact that

(2.4)∫Kφ(

1λx · y) dy =

1n

∫Sn−1

φ?(1λ

(x · v)ρK(v))ρK(v)n dS(v).

To see (2.4) observe that:∫Kφ(

1λx · y) dy =

∫Sn−1

∫ ρK(v)

0φ(

1λ

(x · v)r)rn−1 dr dS(v)

=∫Sn−1

(∫ 1

0φ(

1λ

(x · v)ρK(v)t)tn−1 dt

)ρK(v)ndS(v)

=1n

∫Sn−1

φ?(1λ

(x · v)ρK(v))ρK(v)n dS(v).

Since φ? is strictly increasing on [0,∞) or strictly decreasing on(−∞, 0] it follows that the function

λ 7−→∫Sn−1

φ?( 1λ(x · v)ρK(v)) dV ∗K(v)

is strictly decreasing in (0,∞). It is also continuous. Thus, we have:

Lemma 2.1. Suppose K ∈ Sno and uo ∈ Sn−1. Then∫Sn−1

φ?( 1λo

(uo · v)ρK(v)) dV ∗K(v) = 1

if and only ifhΓφK(uo) = λo.

Observe that (1.4) now shows that Lemma 2.1 holds for all uo ∈Rn \ {0}.

We now demonstrate that (2.1) defines a convex body that containinsthe origin in its interior.

Lemma 2.2. If K ∈ Sno then hΓφK is the support function of a bodyin Kno .

Proof. Observe that it follows immediately from definition (2.1) thatfor x ∈ Rn and c > 0,

hΓφK(cx) = c hΓφK(x).

We show that indeed for x1, x2 ∈ Rn,

hΓφK(x1 + x2) ≤ hΓφK(x1) + hΓφK(x2).

To that end let hΓφK(xi) = λi; i.e.,

(2.5)∫Sn−1

φ?(xi · uλi

ρK(u))dV ∗K(u) = 1.



The convexity of the function s 7→ φ?(s ρK(u)) shows that

φ?(x1 · u+ x2 · uλ1 + λ2

ρK(u))

≤ λ1

λ1 + λ2φ?(x1 · uλ1

ρK(u))

+λ2

λ1 + λ2φ?(x2 · uλ2

ρK(u)).

Integrating both sides of this inequality with respect to the measure V ∗Kand using (2.5) gives∫

Sn−1

φ?(

(x1 + x2) · uλ1 + λ2

ρK(u))dV ∗K(u) ≤ 1,

which, using (2.2), gives the desired result that

hΓφK(x1 + x2) ≤ λ1 + λ2.

Thus hΓφK is indeed the support function of a compact convex set, andsince hΓφK > 0, we see that ΓφK ∈ Kno . q.e.d.

We shall require more than hΓφK > 0. Specifically,

Lemma 2.3. If K ∈ Sno , then

ωn−1rn+1K

ncφ? |K|≤ hΓφK(u) ≤ RK

cφ?,

for all u ∈ Sn−1.

Proof. Suppose uo ∈ Sn−1 and let hΓφK(uo) = λo; i.e.

(2.6)∫Sn−1

φ?(uo · uλo

ρK(u))ρK(u)n dS(u)

n |K|= 1 .

To obtain the lower estimate we proceed as follows. From the defini-tion of cφ? , either φ?(cφ?) = 1 or φ?(−cφ?) = 1. Suppose φ?(cφ?) = 1.Then from the fact that φ? is non-negative and φ?(0) = 0, Jensen’s in-equality, and definition (1.6) together with the fact that φ? is monotoneincreasing on [0,∞) and (1.7),

φ?(cφ?) = 1

=∫Sn−1

φ?(uo · uλo

ρK(u))ρK(u)n dS(u)

n|K|

≥∫Sn−1

φ?(

(uo · u)+

λoρK(u)

)ρK(u)n dS(u)

n|K|

≥ φ?(

1n

∫Sn−1

(uo · u)+

λo

ρK(u)n+1 dS(u)|K|

)≥ φ?

(ωn−1r

n+1K

nλo |K|

).



Since φ? is monotone increasing on [0,∞), from this we obtain the lowerbound for hΓφK :

ωn−1rn+1K

ncφ? |K|≤ λo.

The case where φ?(−cφ?) = 1 is handled the same way and gives thesame result.

To obtain the upper estimate, observe that from the definition ofcφ? , the fact that φ? is monotone decreasing on (−∞, 0] and mono-tone increasing on [0,∞), together with the fact that the functiont 7→ max{φ?(t), φ?(−t)} is monotone increasing on [0,∞), definition(1.6), and finally (2.3) it follows that

max{φ?(cφ?), φ?(−cφ?)}= 1

=∫Sn−1

φ?(uo · uλo

ρK(u))dV ∗K(u)

≤∫Sn−1

max{φ?(|uo · u|ρK(u)

λo

), φ?

(−|uo · u|ρK(u)

λo

)} dV ∗K(u)

≤∫Sn−1

max{φ?(ρK(u)/λo), φ?(−ρK(u)/λo)} dV ∗K(u)

≤∫Sn−1

max{φ?(RK/λo), φ?(−RK/λo)} dV ∗K(u)

= max{φ?(RK/λo), φ?(−RK/λo)}.

But the even function t 7→ max{φ?(t), φ?(−t)} is monotone increasingon [0,∞) so we conclude

λo ≤RKcφ?

.

q.e.d.

For c > 0, an immediate consequence of definition (2.2) and (1.2) isthe fact that

(2.7) ΓφcK = cΓφK.

We next show that the Orlicz centroid operator Γφ : Sno → Sno iscontinuous.

Lemma 2.4. Suppose φ ∈ C. If Ki ∈ Sno and Ki → K ∈ Sno , thenΓφKi → ΓφK.

Proof. Suppose uo ∈ Sn−1. We will show that

hΓφKi(uo)→ hΓφK(uo).

LethΓφKi(uo) = λi,



and note that Lemma 2.3 gives

ωn−1rn+1Ki

ncφ? |Ki|≤ λi ≤

RKicφ?

.

Since Ki → K ∈ Sno , we have rKi → rK > 0 and RKi → RK <∞, andthus there exist a, b such that 0 < a ≤ λi ≤ b < ∞, for all i. To showthat the bounded sequence {λi} converges to hΓφK(uo), we show thatevery convergent subsequence of {λi} converges to hΓφK(uo). Denote anarbitrary convergent subsequence of {λi} by {λi} as well, and supposethat for this subsequence we have

λi → λ∗.

Obviously, a ≤ λ∗ ≤ b. Let Ki = λ−1i Ki. Since λ−1

i → λ−1∗ and

Ki → K, we haveKi → λ−1

∗ K.

Now (2.7), and the fact that hΓφKi(uo) = λi, shows that hΓφKi(uo) = 1;

i.e. ∫Sn−1

φ?((uo · u)ρKi(u)) dV ∗Ki(u) = 1,

for all i. But Ki → λ−1∗ K and the continuity of φ? now give∫

Sn−1

φ?((uo · u)ρλ−1∗ K(u)) dV ∗

λ−1∗ K

(u) = 1,

which by Lemma 2.1 gives

hΓφλ−1∗ K(uo) = 1.

This, (2.7) and (1.4) now give

hΓφK(uo) = λ∗.

This shows that hΓφKi(uo)→ hΓφK(uo) as desired.But for support functions on Sn−1 pointwise and uniform convergence

are equivalent (see, e.g., Schneider [55, p. 54]). Thus, the pointwiseconvergence hΓφKi → hΓφK on Sn−1 completes the proof. q.e.d.

We next show that the Orlicz centroid operator is continuous in φ aswell.

Lemma 2.5. If φi ∈ C and φi → φ ∈ C, then ΓφiK → ΓφK, for eachK ∈ Sno .

Proof. Suppose K ∈ Sno and uo ∈ Sn−1. We will show that for thesupport functions of the convex bodies ΓφiK we have

hΓφiK(uo)→ hΓφK(uo).

LethΓφiK

(uo) = λi,



and note that Lemma 2.3 gives

ωn−1rn+1K

ncφ?i |K|≤ λi ≤

RKcφ?i

.

Since φ?i → φ? ∈ C, we have cφ?i → cφ? ∈ (0,∞) and thus there exista, b such that 0 < a ≤ λi ≤ b <∞, for all i.

To show that the bounded sequence {λi} converges to hΓφK(uo), weshow that every convergent subsequence of {λi} converges to hΓφK(uo).Denote an arbitrary convergent subsequence of {λi} by {λi} as well,and suppose that for this subsequence we have,

λi → λ∗.

Obviously, 0 < a ≤ λ∗ ≤ b. Since h(ΓφiK;uo) = λi,

1 =∫Sn−1

φ?i

(uo · uλi

ρK(u))dV ∗K(u).

This, together with φ?i → φ? ∈ C and λi → λ∗, gives

1 =∫Sn−1

φ?(uo · uλ∗

ρK(u))dV ∗K(u).

But by Lemma 2.1 this gives

hΓφK(uo) = λ∗.

This shows that h(ΓφiK;uo)→ h(ΓφK;uo) as desired.Since the support functions hΓφiK

→ hΓφK pointwise (on Sn−1) theyconverge uniformly and hence

ΓφiK → ΓφK.

q.e.d.

The operator Γφ intertwines with elements of GL(n):

Lemma 2.6. Suppose φ ∈ C. For a star body K ∈ Sno and a lineartransformation A ∈ GL(n),

(2.8) Γφ(AK) = A(ΓφK).

Proof. From (2.7) it follows that we may assume, without loss ofgenerality, that A ∈ SL(n).

Suppose xo ∈ Rn and

h(ΓφAK;xo) = λo.



From Lemma 2.1 and (2.4), the substitution z = Ay and the factsthat |AK| = |K| and dz = |A|dy = dy, we have

1 =1|AK|

∫AK

φ(xo · z/λo) dz

=1|K|

∫Kφ(xo ·Ay/λo) dy

=1|K|

∫Kφ(Atxo · y/λo) dy.

But by Lemma 2.1, (2.4) and (1.5) this implies that

λo = h(ΓφK;Atxo) = h(AΓφK;xo),giving h(ΓφAK;xo) = h(AΓφK;xo). q.e.d.

3. Proof of the Orlicz Busemann-Petty centroid inequality

The proof of our theorem makes critical use of:

Lemma 3.1. Suppose φ ∈ C, and K ∈ Kno . If u ∈ Sn−1 and x′1, x′2 ∈

u⊥, then

(3.1) h(Γφ(SuK); 12x′1 + 1

2x′2, 1) ≤ 1

2h(ΓφK;x′1, 1) +

12h(ΓφK;x′2,−1).

Equality in the inequality implies that all of the chords of K parallel tou, whose distance from the origin is less than

rK2 max{1, |x′1|, |x′2|}

,

have midpoints that lie in a subspace.

Proof. In light of Lemma 2.6 we may assume, without loss of gener-ality, that |K| = 1 = |SuK|.

Let K ′ = Ku denote the image of the projection of K onto the sub-space u⊥. For each y′ ∈ K ′, let σy′(u) = σy′ = |K ∩ (y′ + Ru)| be thelength of the chord K ∩ (y′+ Ru), and let my′ = my′(u) be defined suchthat y′ +my′u is the midpoint of the chord K ∩ (y′ + Ru).

For λ1, λ2 > 0, we have∫Kφ

((x′1, 1) · y

λ1

)dy =

∫Kφ

((x′1, 1) · (y′, s)

λ1

)dy′ds

=∫K′dy′∫ my′+σy′/2

my′−σy′/2φ

(x′1 · y′ + s

λ1

)ds

=∫K′dy′∫ σy′/2

−σy′/2φ

(x′1 · y′ + t+my′

λ1

)dt

=∫

SuKφ

(x′1 · y′ + t+my′(u)

λ1

)dy′dt,

(3.2)



by making the change of variables t = −my′ + s, and

∫Kφ

((x′2,−1) · y

λ2

)dy =

∫Kφ

((x′2,−1) · (y′, s)

λ2

)dy′ds

=∫K′dy′∫ my′+σy′/2

my′−σy′/2φ

(x′2 · y′ − s

λ2

)ds

=∫K′dy′∫ σy′/2

−σy′/2φ

(x′2 · y′ + t−my′

λ2

)dt

=∫

SuKφ

(x′2 · y′ + t−my′(u)

λ2

)dy′dt,

(3.3)

by making the change of variables t = my′ − s.Abbreviate

x′o = 12x′1 + 1

2x′2 and λo = 1

2λ1 + 12λ2,

and from the convexity of φ, follows(3.4)

2φ(x′o ·y′ + t

λo

)≤ λ1

λoφ

(x′1 ·y′ + t+my′

λ1

)+λ2

λoφ

(x′2 ·y′ + t−my′

λ2

).

From (3.2) - (3.4), we have

λ1

λo

∫Kφ

((x′1, 1) · y

λ1

)dy +

λ2

λo

∫Kφ

((x′2,−1) · y

λ2

)dy

=λ1

λo

∫SuK

φ

(x′1 · y′ + t+my′(u)

λ1

)dy′dt

+λ2

λo

∫SuK

φ

(x′2 · y′ + t−my′(u)

λ2

)dy′dt

≥ 2∫

SuKφ

((1

2x′1 + 1

2x′2) · y′ + t

12λ1 + 1

2λ2

)dy′dt

= 2∫

SuKφ

((1

2x′1 + 1

2x′2, 1) · (y′, t)

12λ1 + 1

2λ2

)dy′dt

= 2∫

SuKφ

((x′o, 1) · y

λo

)dy.

(3.5)

Choose

λ1 = h(ΓφK;x′1, 1) and λ2 = h(ΓφK;x′2,−1);

recall that |K| = 1 and we have from Lemma 2.1,∫Kφ

((x′1, 1) · y

λ1

)dy = 1 and

∫Kφ

((x′2,−1) · y

λ2

)dy = 1.



But this in (3.5), and the fact that | SuK| = 1 shows that

1 ≥ 1|SuK|

∫SuK

φ

((1

2x′1 + 1

2x′2, 1) · y

12λ1 + 1

2λ2

)dy,

which by definition (2.1) gives

h(Γφ(SuK); 12x′1 + 1

2x′2, 1) ≤ 1

2λ1 + 12λ2,

with equality forcing (in light of the continuity of φ) equality in (3.4)for all y′ ∈ K ′ and all t ∈ [−σy′/2, σy′/2].

This establishes the desired inequality.Suppose there is equality. Hence there is equality in (3.4) for all

y′ ∈ K ′ and all t ∈ [−σy′/2, σy′/2].From definition (1.6) of rK we see that if | y′| < rK/2 then

(−rK2,rK2

) ⊂ (my′ −σy′

2,my′ +

σy′

2)(3.6a)

and therefore also

(−rK2,rK2

) ⊂ (−my′ −σy′

2,−my′ +

σy′

2).(3.6b)

Suppose y′ is such that

| y′| < rK2 max{1, |x′1|, |x′2|}

.

Then,

x′1 · y′ ∈ (−rK2,rK2

) and x′2 · y′ ∈ (−rK2,rK2

)

and from (3.6) it follows that

x′1 · y′ +my′ ∈ (−σy′

2,σy′

2) and x′2 · y′ −my′ ∈ (−

σy′

2,σy′

2).

Thus, the linear functions

t 7→ x′1 · y′ + t+my′ and t 7→ x′2 · y′ + t−my′

both have their root in (−σy′/2, σy′/2). Thus, they either (1) have theirroot at the same t = ty′ ∈ (−σy′/2, σy′/2) or (2) there will exist at = t?y′ ∈ (−σy′/2, σy′/2) at which these functions have opposite signs.

Consider case (2) first. The fact that

x′1 · y′ + t?y′ +my′ and x′2 · y′ + t?y′ −my′

have opposite signs tells us that

x′1 · y′ + t+my′ and x′2 · y′ + t−my′



have opposite signs for all t ∈ (t?y′ − δy′ , t?y′ + δy′) for some δy′ > 0. Thisand the fact that there is equality in (3.4) together with the fact thatφ can not be linear in a neighborhood of the origin gives

x′1 ·y′ + t+my′

λ1=x′2 ·y′ + t−my′

λ2,

for all t ∈ (t?y′ − δy′ , t?y′ + δy′) which contradicts the assumption that thelinear functions have opposite signs.

In case (1) the linear functions

t 7→ x′1 · y′ + t+my′ and t 7→ x′2 · y′ + t−my′

have a root at the same t = ty′ ∈ (−σy′/2, σy′/2) and this immediatelyyields

(x′2 − x′1) · y′ = 2my′ .

But this means that for | y′| < rK/max{2, 2|x′1|, 2|x′2|}, the midpoints

{(y′,my′) : y′ ∈ K ′}

of the chords of K parallel to u lie in the subspace

{(y′, 12(x′2 − x′1) · y′) : y′ ∈ K ′}

of Rn. q.e.d.

As an aside, observe that the inequality of Lemma 3.1 could havebeen presented as:

h(Γφ(SuK); 12x′1 + 1

2x′2,−1) ≤ 1

2h(ΓφK;x′1, 1) +

12h(ΓφK;x′2,−1).

If φ is assumed to be strictly convex, then the equality conditions ofthe inequality in Lemma 3.1 are simple.

Lemma 3.2. Suppose φ is strictly convex and K ∈ Kno . If u ∈ Sn−1

and x′1, x′2 ∈ u⊥, then

h(Γφ(SuK); 12x′1 + 1

2x′2, 1) ≤ 1

2h(ΓφK;x′1, 1) +

12h(ΓφK;x′2,−1),

and

h(Γφ(SuK); 12x′1 + 1

2x′2,−1) ≤ 1

2h(ΓφK;x′1, 1) +

12h(ΓφK;x′2,−1).

Equality in either inequality, implies

h(ΓφK;x′1, 1) = h(ΓφK;x′2,−1)

and that all of the midpoints of the chords of K parallel to u lie in asubspace.



Proof. Observe that equality forces equality in (3.4) for all y′ ∈ K ′and all t ∈ [−σy′/2, σy′/2]. But since φ is strictly convex this meansthat we must have

(3.7)x′1 ·y′ + t+my′

λ1=x′2 ·y′ + t−my′

λ2,

for all t ∈ (−σy′/2, σy′/2). Choosing λ1 = h(ΓφK;x′1, 1) and λ2 =h(ΓφK;x′2,−1), (3.7) immediately gives

h(ΓφK;x′1, 1) = λ1 = λ2 = h(ΓφK;x′2,−1),

and

(x′2 − x′1) · y′ = 2my′ ,

for all y′ ∈ K ′. But this means that the midpoints {(y′,my′) : y′ ∈ K ′}of the chords of K parallel to u lie in the subspace

{(y′, 12

(x′2 − x′1) · y′) : y′ ∈ K ′}

of Rn. q.e.d.

The theorem will be proved using:

Lemma 3.3. Suppose φ ∈ C and K ∈ Kno . If u ∈ Sn−1, then

(3.8) Γφ(SuK) ⊂ Su(ΓφK).

If the inclusion is an identity then all of the chords of K parallel to u,whose distance from the origin is less than

rK rΓφK

4RΓφK,

have midpoints that lie in a subspace.

Proof. Suppose y′ ∈ relint (ΓφK)u. By Lemma 1.2 there exist x′1 =x′1(y′) and x′2 = x′2(y′) in u⊥ such that

`u(ΓφK, y′) = hΓφK(x′1, 1)− x′1 · y′,`u(ΓφK, y′) = hΓφK(x′2,−1)− x′2 · y′.

(3.9)



Now by (1.8), (3.9), followed by Lemma 3.1, and then Lemma 1.2 wehave

ù(Su(ΓφK); y′) =12ù(ΓφK; y′) +

12ù(ΓφK; y′)

=12(hΓφK(x′1, 1)− x′1 · y′

)+

12(hΓφK(x′2,−1)− x′2 · y′

)=

12hΓφK(x′1, 1) +

12hΓφK(x′2,−1)− (1

2x′1 + 1

2x′2) · y′

≥ hΓφ(SuK)(12x′1 + 1

2x′2, 1)− (1

2x′1 + 1

2x′2) · y′

≥ minx′∈u⊥

{hΓφ(SuK)(x

′, 1)− x′ · y′}

= ù(Γφ(SuK); y′),

(3.10)

and

ù(Su(ΓφK); y′) =12ù(ΓφK; y′) +

12ù(ΓφK; y′)

=12(hΓφK(x′1, 1)− x′1 · y′

)+

12(hΓφK(x′2,−1)− x′2 · y′

)=

12hΓφK(x′1, 1) +

12hΓφK(x′2,−1)− (1

2x′1 + 1

2x′2) · y′

≥ hΓφ(SuK)(12x′1 + 1

2x′2,−1)− (1

2x′1 + 1

2x′2) · y′

≥ minx′∈u⊥

{hΓφ(SuK)(x

′,−1)− x′ · y′}

= ù(Γφ(SuK); y′).

This establishes the inclusion.Now suppose

Γφ(SuK) = Su(ΓφK).Then by Lemma 1.2, for each y′ ∈ (ΓφK)u ∩ (rΓφK/2)B, there exist

x′1 = x′1(y′) and x′2 = x′2(y′) in u⊥ such that

ù(ΓφK, y′) = hΓφK(x′1, 1)− x′1 · y′,ù(ΓφK, y′) = hΓφK(x′2,−1)− x′2 · y′,

(3.11)

and since Γφ(SuK) = Su(ΓφK), from (3.10) we see that

(3.12) hΓφ(SuK)(12x′1 + 1

2x′2, 1) =

12hΓφK(x′1, 1) +

12hΓφK(x′2,−1).

From Lemma 1.3 and (3.11), it follows that both

|x′1|, |x′2| ≤2RΓφK

rΓφK.

But now (3.12) and the equality conditions of Lemma 3.1 show that allof the chords of K parallel to u, whose distance from the origin is less



thanrK rΓφK

4RΓφK,

have midpoints that lie in a subspace. q.e.d.

As a direct consequence of Lemma 1.1, we now have:

Corollary. Suppose φ ∈ C and K ∈ Kno . If u ∈ Sn−1, then

Γφ(SuK) ⊂ Su(ΓφK).

If the inclusion is an identity for all u, then K is an ellipsoid centeredat the origin.

The Corollary and a standard Steiner symmetrization argument nowyield:

Theorem. If φ ∈ C and K ∈ Kno , then the volume ratio |ΓφK|/|K|is minimized if and only if K is an ellipsoid centered at the origin.

4. Open Problems

The class-reduction technique introduced in [33] can be used to showthat once the Busemann-Petty centroid inequality and its equality con-ditions have been established for convex bodies (in fact for a muchsmaller class of bodies) then one can easily extend the inequality andits equality conditions to all star bodies. It was shown in [37] thatthis is also the case for Lp centroid bodies. Does there exist a similarclass-reduction technique that is applicable for Orlicz centroid bodies?

Conjecture. If φ ∈ C and K ∈ Sno , then the volume ratio |ΓφK|/|K|is minimized only by ellipsoids.

In [45], the Orlicz projection body ΠφK of a convex body K ∈ Kno ,was defined as the convex body whose support function is given by

(4.1) hΠφK(x) = inf{λ > 0 :

∫Sn−1

φ( 1λ(x · u)ρK∗(u)) dVK(u) ≤ 1

},

where VK , the volume-normalized conical measure, is defined by

|K|dVK =1nhK dSK ,

and SK is the classical Aleksandrov-Fenchel-Jessen surface area measureof K. Let Π∗φK = (ΠφK)∗ denote the polar Orlicz projection body ofK. Compare definition (4.1) with definition (2.2).

In [45], the following inequality was established.

Orlicz Petty projection inequality. Suppose φ ∈ C is strictlyconvex. If K ∈ Kno then the volume ratio

|Π∗φK|/|K|is maximal if and only if K is an ellipsoid centered at the origin.



The technique introduced in [33] shows that once the Petty Pro-jection inequality has been established then one can easily derive theBusemann-Petty centroid inequality as a consequence, and vice versa.The same is true of the Lp Busemann-Petty centroid inequality andthe Lp Petty projection inequality. Is there an easy road from the Or-licz Petty projection inequality to the Orlicz Busemann-Petty centroidinequality? Is there an easy road from the Orlicz Busemann-Petty cen-troid inequality to the Orlicz Petty projection inequality?

5. Acknowledgement

The authors are indebted to the referees for their very careful readingof the original submission.

References

[1] J. Bastero and M. Romance, Positions of convex bodies associated to ex-tremal problems and isotropic measures, Adv. Math. 184 (2004), no. 1, 64–88,MR2047849, Zbl 1053.52011.

[2] C. Bianchini and A. Colesanti, A sharp Rogers and Shephard inequality for thep-difference body of planar convex bodies, Proc. Amer. Math. Soc. 136 (2008),2575–2582, MR2390529, Zbl 1143.52008.

[3] S. Campi and P. Gronchi, The Lp-Busemann-Petty centroid inequality, Adv.Math. 167 (2002), 128–141, MR1901248, Zbl 1002.52005.

[4] , On the reverse Lp-Busemann-Petty centroid inequality, Mathematika49 (2002), 1–11, MR2059037, Zbl 1056.52005.

[5] , Extremal convex sets for Sylvester-Busemann type functionals, Appl.Anal. 85 (2006), 129–141, MR2198835, Zbl 1090.52006.

[6] , On volume product inequalities for convex sets, Proc. Amer. Math. Soc.134 (2006) 2393–2402, MR2213713, Zbl 1095.52002.

[7] , Volume inequalities for Lp-zonotopes, Mathematika 53 (2006), 71–80(2007), MR2304053, Zbl 1117.52011.

[8] A. Cianchi, E. Lutwak, D. Yang, and G. Zhang, Affine Moser-Trudinger andMorrey-Sobolev inequalities, Calc. Var. Partial Differential Equations (in press).

[9] K. S. Chou and X. J. Wang, The Lp-Minkowski problem and the Minkowskiproblem in centroaffine geometry, Adv. Math. 205 2006, 33–83, MR2254308,Zbl pre05054348.

[10] N. Dafnis and G. Paouris, Small ball probability estimates, Ψ2-behavior and thehyperplane conjecture, J. Funct. Anal. (in press).

[11] B. Fleury, O. Guedon, and G. A. Paouris A stability result for mean width of Lp-centroid bodies, Adv. Math. 214 (2007), 865–877, MR2349721, Zbl 1132.52012

[12] R. J. Gardner, Geometric Tomography, Second edition. Encyclopedia of Mathe-matics and its Applications, 58. Cambridge University Press, New York, 2006,MR2251886, Zbl 1102.52002.

[13] , The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39(2002), 355–405, MR1898210, Zbl 1019.26008.



[14] A. Giannopoulos, A. Pajor, and G. Paouris, A note on subgaussian estimates forlinear functionals on convex bodies, Proc. Amer. Math. Soc. 135 (2007) 2599–2606, MR2302581, Zbl 1120.52003.

[15] P. M. Gruber, A note on ellipsoids (manuscript).

[16] P. M. Gruber and M. Ludwig, A Helmholtz-Lie type characterization of ellipsoidsII, Discrete Comput. Geom. 16 (1996), 55–67. MR1397787 (97h:52002), Zbl0864.52005.

[17] P. Guan and C.-S. Lin, On equation det(uij + δiju) = upf on Sn, preprint.

[18] C. Haberl, Lp intersection bodies, Adv. Math. 217, (2008) 2599-2624,MR2397461, Zbl 1140.52003.

[19] C. Haberl, Star body valued valuations, Indiana Univ. Math. J. (in press).

[20] C. Haberl and M. Ludwig, A characterization of Lp intersection bodies, Int.Math. Res. Not. 2006, No. 17, Article ID 10548, 29 p. (2006), MR2250020, Zbl1115.52006.

[21] C. Haberl and F. Schuster, General Lp affine isoperimetric inequalities, J. Dif-ferential Geom. (in press).

[22] C. Haberl and F. Schuster, Asymmetric affine Lp Sobolev inequalities, J. Funct.Anal. 257, 2009, 641-658.

[23] C. Hu, X.-N. Ma, and C. Shen, On the Christoffel-Minkowski problem ofFirey’s p-sum, Calc. Var. Partial Differential Equations 21 (2004), 137–155,MR2085300, Zbl pre02113880.

[24] R. Lata la, J. O. Wojtaszczyk, On the infimum convolution inequality, StudiaMath. 189 (2008), 147–187, MR2449135, Zbl pre05354528.

[25] K. Leichtweiss, Affine geometry of convex bodies, Johann Ambrosius Barth Ver-lag, Heidelberg, 1998, MR1630116, Zbl 0899.52005.

[26] M. Ludwig, Projection bodies and valuations, Adv. Math. 172 (2002), 158–168,MR1942402, Zbl 1019.52003.

[27] , Valuations on polytopes containing the origin in their interiors, Adv.Math. 170 (2002), 239–256, MR1932331, Zbl 1015.52012.

[28] , Ellipsoids and matrix-valued valuations, Duke Math. J. 119 (2003),159–188, MR1991649, Zbl 1033.52012.

[29] , Minkowski valuations, Trans. Amer. Math. Soc. 357 (2005), 4191–4213,MR2159706, Zbl 1077.52005.

[30] , Intersection bodies and valuations, Amer. J. Math. 128 (2006), 1409–1428, MR2275906, Zbl 1115.52007.

[31] , General affine surface areas, Comment. Math. Helv. (in press).

[32] M. Ludwig and M. Reitzner. A classification of SL(n) invariant valuations, Ann.of Math. (in press).

[33] E. Lutwak, On some affine isoperimetric inequalities, J. Differential Geom. 23,(1986) 1-13, MR0840399, Zbl 0592.52005.

[34] , The Brunn-Minkowski-Firey theory. I. Mixed volumes and theMinkowski problem, J. Differential Geom. 38 (1993), 131–150, MR1231704, Zbl0788.52007.

[35] , The Brunn-Minkowski-Firey theory. II. Affine and geominimal surfaceareas, Adv. Math. 118 (1996), 244–294, MR1378681, Zbl 0853.52005.



[36] E. Lutwak and V. Oliker, On the regularity of solutions to a generalization ofthe Minkowski problem, J. Differential Geom. 41 (1995), 227–246, MR1316557,Zbl 0867.52003.

[37] E. Lutwak, D. Yang, and G. Zhang, Lp affine isoperimetric inequalities, J. Dif-ferential Geom. 56 (2000), 111–132, MR1863023, Zbl 1034.52009.

[38] , A new ellipsoid associated with convex bodies, Duke Math. J. 104 (2000),375–390, MR1781476, Zbl 0974.52008.

[39] , The Cramer-Rao inequality for star bodies, Duke Math. J. 112 (2002),59–81, MR1890647, Zbl 1021.52008.

[40] , Sharp affine Lp Sobolev inequalities, J. Differential Geom. 62 (2002),17–38, MR1987375, Zbl 1073.46027.

[41] , Volume inequalities for subspaces of Lp, J. Differential Geom. 68 (2004),159–184, MR2152912, Zbl 1119.52006.

[42] , Lp John ellipsoids, Proc. London Math. Soc. 90(2005), 497–520,MR2142136, Zbl 1074.52005.

[43] , Optimal Sobolev norms and the Lp Minkowski problem, Int. Math. Res.Not. (2006), Article ID 62987, 1–21, MR2211138, Zbl 1110.46023.

[44] , Volume inequalities for isotropic measures, Amer. J. Math. 129 (2007),1711–1723, MR2369894, Zbl 1134.52010.

[45] , Orlicz projection bodies, Adv. Math. (in press).

[46] E. Lutwak and G. Zhang, Blaschke-Santalo inequalities, J. Differential Geom.47 (1997), 1–16, MR1601426, Zbl 0906.52003.

[47] M. Meyer and E. Werner, On the p-affine surface area, Adv. Math. 152 (2000),288–313, MR1764106, Zbl 0964.52005.

[48] I. Molchanov, Convex and star-shaped sets associated with multivariate stabledistributions. I. Moments and densities, J. Multivariate Anal. (in press).

[49] G. Paouris, On the ψ2-behaviour of linear functionals on isotropic convex bodies,Stud. Math. 168, (2005) 285-299 (2005), MR2146128, Zbl 1078.52501.

[50] , Concentration of mass on convex bodies, Geom. Funct. Anal. 16 (2006),1021–1049, MR2276533, Zbl 1114.52004.

[51] , Concentration of mass on isotropic convex bodies, C. R. Math. Acad.Sci. Paris 342 (2006), 179–182.

[52] , Small ball probability estimates for log-concave measures, (preprint).

[53] C. M. Petty, Centroid surfaces, Pac. J. Math. 11 (1961) 1535-1547, MR0133733,Zbl 0103.15604.

[54] D. Ryabogin and A. Zvavitch, The Fourier transform and Firey projections ofconvex bodies, Indiana Univ. Math. J. 53 (2004), 667–682, MR2086696, Zbl1062.52004.

[55] R. Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia ofMathematics and its Applications, vol. 44, Cambridge University Press, Cam-bridge, 1993, MR1216521, Zbl 0798.52001.

[56] C. Schutt and E. Werner, Surface bodies and p-affine surface area, Adv. Math.187 (2004), 98–145, MR2074173, Zbl 1089.52002.

[57] A. Stancu, The discrete planar L0-Minkowski problem, Adv. Math. 167 (2002),160–174, MR1901250, Zbl 1005.52002.



[58] A. C. Thompson, Minkowski geometry, Encyclopedia of Mathematics and its Ap-plications, vol. 63, Cambridge University Press, Cambridge, 1996, MR1406315,Zbl 0868.52001.

[59] V. Umanskiy, On solvability of two-dimensional Lp-Minkowski problem, Adv.Math. 180 (2003), 176–186, MR2019221, Zbl 1048.52001.

[60] E. Werner, On Lp-affine surface areas, Indiana Univ. Math. J. 56 (2007), 2305–2323, MR2360611, Zbl 1132.52008.

[61] E. Werner and D.-P. Ye, New Lp affine isoperimetric inequalities, Adv. Math.218 (2008), 762–780, MR2414321, Zbl 1155.52002.

[62] V. Yaskin and M. Yaskina, Centroid bodies and comparison of volumes, IndianaUniv. Math. J. 55 (2006), 1175–1194, MR2244603, Zbl 1102.52005.

[63] G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), 183–202, MR1776095, Zbl 1040.53089.

Polytechnic Institute of NYUSix MetroTech Center

Brooklyn NY 11201


orlicz centroid bodies erwin lutwak, deane yang and

Documents