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VOLUME INEQUALITIES FOR SECTIONS AND PROJECTIONS OF WULFF SHAPES AND THEIR POLARS

AI-JUN LI, QINGZHONG HUANG, AND DONGMENG XI

Abstract. Let 1kn. Sharp volume inequalities fork-dimensional sections of Wulff shapes and dual inequalities for projections are established. As their applications, several special Wulff shapes are investigated.

1. Introduction

Throughout, all Borel measures are understood to be nonnegative and finite. A convex body in Rnis a compact convex set containing the origin in its interior. The polar body of a convex body K is given by K = {x Rn : x·y 1 for all y K}, where x·y denotes the standard inner product of x and y in Rn. We use k · k to denote the Euclidean norm on Rn. When A is a compact convex set in Rn, we write |A| for the volume of A in the appropriate subspace. Let suppν denote the support of a measure ν and let PH be the orthogonal projection onto a subspace H of Rn.

Volume estimates for sections of convex bodies inRn are not easy, even in specific cases. For the cubeQn= [−12,12]n, Hensley [13] first showed that ifHis a hyperplane of Rn then |H ∩Qn| lies between 1 and 5, and conjectured that the upper bound is at most

2. This conjecture was solved by Ball [1, 2], who also settled the more general case of k-dimensional sections.

2000Mathematics Subject Classification. 52A40.

Key words and phrases. Wulff shape, isotropic measure, support subset.

The first author was supported by NSFC-Henan Joint Fund (No. U1204102) and Key Re- search Project for Higher Education in Henan Province (No. 17A110022). The second author was supported in part by the National Natural Science Foundation of China (No. 11626115 and 11371239). The third author was supported by the National Natural Science Foundation of China (No. 11601310) and Shanghai Sailing Program (No. 16YF1403800).

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The example of the regular simplex is much more complicated. Webb [30] proved that the maximal central hyperplane section is the one containingn−1 vertices and the centroid. The question about the minimal central hyperplane section has not been completely solved yet. Brzezinski [10] proved a lower bound which differs from the conjectured minimal volume by a factor of approximately 1.27. For general k- dimensional sections, these questions were recently considered by Dirksen [11]. Other examples, such as `np-balls [7, 9, 25], complex cubes [27] and non-central sections of cubes [26], have also been investigated.

In this paper, we will study sections and projections of more general convex bodies than cubes and simplices. The main objects we consider are Wulff shapes [28], which were introduced by Wulff in 1901. Nowadays, it is an important notion in convex geometric analysis (see, e.g., [28]).

Definition. Suppose that ν is a Borel measure on Sn−1 and that f is a positive, bounded, and measurable function on Sn−1. The Wulff shape Wν,f determined by ν and f is defined by

Wν,f :={x∈Rn :x·u≤f(u) for all u∈suppν}. (1.1) The measure ν is said to be even if it assumes the same value on antipodal sets.

When ν and f are both even, then Wν,f is origin-symmetric. It is easy to see that Wν,f is always convex and may be unbounded. In order for ak-dimensional subspace H ofRn, to guarantee that|Wν,f|and |H∩Wν,f|are finite, we consider Wulff shapes determined by measuresν which are isotropic and f-centered with respect to H. A Borel measure ν onSn−1 is called isotropic if

Z

Sn−1

u⊗udν(u) = In, (1.2)

whereu⊗uis the rank-one orthogonal projection onto the space spanned by uand In is the identity map on Rn. The definition that the measure ν isf-centered with respect to H needs more description.

LetH be a k-dimensional subspace of Rn with 1 ≤k ≤n and let ¯ν be the Borel measure on Sn−1∩H defined by

¯ ν(A) =

Z

Sn−1\H

1A

³ PHu kPHuk

´

kPHuk2dν(u) (1.3)

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for Borel sets A Sn−1 ∩H. It is easy to see that the support set of ¯ν is exactly the set of {PHu/kPHuk:u suppν\H}. The section H∩Wν,f can be expressed as

H∩Wν,f ={y∈H :y·u≤f(u) for all u∈suppν}

= n

y∈H : PHu

kPHuk f(u)

kPHuk for all u∈suppν\H o

= n

y∈H :y·w≤ inf

u∈Ξw

f(u)

kPHuk for all w∈supp ¯ν o

, (1.4)

where

Ξw = n

u∈suppν\H : PHu

kPHuk =w, w supp ¯ν o

. The measureν is calledf-centered with respect to H if

Z

ΨH

f(u)PHudν(u) = o, (1.5) where ΨH suppν is the support subset (with respect to H) ofν, defined by

ΨH = [

w∈supp ¯ν

Ψw, where

Ψw = n

uw Ξw : f(uw)

kPHuwk = inf

u∈Ξw

f(u) kPHuk

o .

Sincef is measurable and PH is continuous, Ψw is a measurable set in suppν\H. Obviously, if supp ¯ν is countable or finite, then ΨH is a measurable set. So, when 1 k < n, we always assume that the measure ν, as well as ¯ν, is discrete in our main results (Theorems 1.1 and 1.2) to guarantee that ΨH is measurable. If k =n, then H = Rn, PH = In and thus, ΨH = ΨRn = suppν, a measurable set. In this case, (1.5) reduces to Z

Sn−1

f(u)udν(u) = o, (1.6)

and the measure ν is f-centered (see e.g., [29]). Note that the regular simplex (and the cube) is a Wulff shape determined by a (even) discrete measure ν which is f-centered and isotropic.

We also need the notion of displacement of H∩Wν,f defined by disp (H∩Wν,f) = 1

|H∩Wν,f| Z

H∩Wν,f

xdx· Z

ΨH

PHu

f(u)kPHuk2dν(u), (1.7)

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and

kfkL2H) =

³ Z

ΨH

f(u)2dν(u)

´1/2

. (1.8)

The aim of this paper is to establish volume inequalities for sections and pro- jections of Wulff shapes and their polars. For the cases 1 k < n we make the additional assumption of discreteness of the underlying isotropic measureν to guar- antee that ΨH is measurable. When k = n, the set ΨH = suppν is measurable.

Then the assumption of discreteness of ν is unnecessary and we recover the results by Schuster and Weberndorfer [29], and generalized and unified results of Ball [2,3], Barthe [6], and Lutwak, Yang, and Zhang [22, 23].

The asymmetric case can be stated as follows:

Theorem 1.1. Let H be a k-dimensional subspace of Rn with 1≤k≤ n and let f be a positive bounded measurable function on Sn−1. Suppose that the measure ν on Sn−1 is discrete, isotropic, and f-centered with respect to H when 1 k < n, and that ν is isotropic and f-centered when k =n. Then

|H∩Wν,f| ≤ (k+ 1disp(H∩Wν,f))k+1

k!(k+ 1)k+12 kfkkL2H), and

|PHWν,f | ≥ (k+ 1)k+12

k! kfk−kL2

H),

with equalities if and only ifH∩Wν,f is a regular simplex inscribed in kfkL2(ΨHk )(Sn−1 H) and kPf(u)

Huk is constant for all u∈ΨH. The symmetric case reads as follows:

Theorem 1.2. Let H be a k-dimensional subspace of Rn with 1≤k≤ n and let f be an even positive bounded measurable function on Sn−1. Suppose that the measure ν onSn−1 is discrete and even isotropic when1≤k < n, and thatν is even isotropic when k =n. Then

|H∩Wν,f| ≤

³ 2

√k

´k

kfkkL2H), and

|PHWν,f | ≥ (2 k)k

(k)! kfk−kL2H),

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with equalities if and only if H∩Wν,f is an origin-symmetric cube with side length of 2kkfkL2H) and kPf(u)

Huk is constant for all u∈ΨH.

Our proofs are based on a refinement of the approaches taken by Schuster and Weberndorfer [29], Lutwak, Yang, and Zhang [23], and Ball [3]. More precisely, the concept of isotropic embedding and the Ball-Barthe inequality play important roles in the proofs. For more applications of this approach, see e.g., [14, 16–19, 21, 22, 29].

This paper is organized as follows. In Section 2, background material is pro- vided. Sections 3 and 4 contain the proofs of Theorem 1.1 and Theorem 1.2. As applications, several special Wulff shapes are investigated in Section 5.

2. Background material

We collect some background material. General references are the books of Gardner [12] and Schneider [28].

Lete1, . . . , enbe the standard orthonormal basis ofRn. The Minkowski functional k · kK of a convex body K inRn is defined by

kxkK = min{t >0 :x∈tK}.

By using polar coordinates, we have

|K|= 1 Γ(1 + np)

Z

Rn

e−kxkpKdx, (2.1)

where integration is with respect to Lebesgue measure on Rn. It is well-known that the isotropy condition (1.2) is equivalent to

kxk2 = Z

Sn−1

|x·u|2dν(u), (2.2) for allx∈Rn. Special cases of even isotropic measures are thecross measures, which are concentrated uniformly on {±u1, . . . ,±un}, where u1, . . . , un is an orthonormal basis of Rn.

Suppose that H is a k-dimensional subspace of Rn with 1 k n. Let ν be a Borel measure onSn−1 and let ¯ν be the Borel measure onSn−1∩H defined in (1.3).

Then for any continuous functiong :Sn−1∩H R, Z

Sn−1∩H

g(w)d¯ν(w) = Z

Sn−1\H

g

³ PHu kPHuk

´

kPHuk2dν(u). (2.3)

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If ν is an isotropic measure on Sn−1, then for an arbitraryy ∈H, we have Z

Sn−1∩H

|y·w|2d¯ν(w) = Z

Sn−1\H

¯¯

¯y· PHu kPHuk

¯¯

¯2kPHuk2dν(u)

= Z

Sn−1

|y·u|2dν(u) =kyk2. Hence, ¯ν is isotropic onSn−1 ∩H. Moreover, we have

¯

ν(Sn−1∩H) =k. (2.4)

Letf be a positive, bounded, and measurable function on Sn−1. By (1.4), we have H∩Wν,f =n

y∈H :y·w≤ inf

u∈Ξw

f(u)

kPHuk for all w∈supp ¯νo

={y∈H :y·w≤f¯(w) for all w∈supp ¯ν}

:= ¯W¯ν,f¯, (2.5)

where the function ¯f :Sn−1∩H→(0,∞) is defined by, forw∈supp ¯ν, f(w) = inf¯

u∈Ξw

f(u)

kPHuk = f(u)

kPHuk, u∈Ψw. (2.6) For a k-dimensional subspace H, define the (k + 1)-dimensional subspace H0 in Rn+1 =Rn×Rby

H0 = span{H, en+1}. (2.7)

Since we can identify Sn−1∩H with Sk−1 inRk, and identifySn∩H0 with Sk in Rk+1, a lemma due to Schuster and Weberndorfer [29, Lemma 4.1] can be restated as follows.

Lemma 2.1. Let ν¯ be an isotropic measure on Sn−1 ∩H and let f¯be a positive bounded measurable function on Sn−1∩H. For w∈Sn−1∩H, define the functions ϕ± :Sn−1∩H →H0\{0} by

ϕ±(w) = (±w,f¯(w)). (2.8)

Then the measure ν˜ on Sn∩H0, defined by Z

Sn∩H0

˜

g(η)dν(η) =˜ Z

Sn−1∩H

˜ g

³ ϕ±(w) ±(w)k

´

±(w)k2d¯ν(w) (2.9) for every continuous ˜g : Sn ∩H0 R, is isotropic if and only if ν¯ is f¯-centered on H and kf¯kL2ν)= 1.

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Note that the functionsϕ±lift the isotropic measure ¯νonSn−1∩H to the isotropic measure ˜ν on Sn ∩H0 and are therefore usually called isotropic embeddings of ¯ν.

They were introduced by Lutwak, Yang, and Zhang [23].

Lemma 2.2. If the measure ν on Sn−1 isf-centered with respect toH and the sup- port subset ΨH of ν is measurable such that kfkL2H) = 1, then the measure ν˜ de- fined in (2.9) is isotropic on Sn∩H0.

Proof. By Lemma 2.1, it is sufficient to verify that ¯νis ¯f-centered onHandkf¯kL2ν)= 1. Indeed, it follows from (2.3), (2.6), and (1.5) that

Z

Sn−1∩H

f¯(w)wd¯ν(w) = Z

Sn−1\H

f¯³ PHu kPHuk

´ PHu

kPHukkPHuk2dν(u)

= Z

ΨH

f(u) kPHuk

PHu

kPHukkPHuk2dν(u)

= Z

ΨH

f(u)PHudν(u) =o. (2.10) That is, ¯ν is ¯f-centered on H. Moreover, by (2.3) and (2.6), we have

kfk¯ L2ν) =

³ Z

Sn−1∩H

f(w)¯ 2d¯ν(w)

´1/2

=³ Z

Sn−1\H

f¯³ PHu kPHuk

´2

kPHuk2dν(u)´1/2

=

³ Z

ΨH

³ f(u) kPHuk

´2

kPHuk2dν(u)

´1/2

=kfkL2H) = 1, (2.11)

which concludes the proof. ¤

Define the displacement of ¯Wν,¯f¯ by disp( ¯Wν,¯f¯) = 1

|W¯ν,¯f¯| Z

W¯ν,¯f¯

xdx· Z

Sn−1∩H

w

f(w)¯ d¯ν(w). (2.12) Then, it follows from (2.5), (2.3), and (1.7) that

disp(H∩Wν,f) = disp( ¯W¯ν,f¯). (2.13)

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The following continuous version of the Ball-Barthe inequality was established by Lutwak, Yang, and Zhang [21], extending the discrete case due to Ball and Barthe [6].

Lemma 2.3. If ν is an isotropic measure on Sk−1 in Rk and t is a positive contin- uous function on suppν, then

det Z

Sk−1

t(u)u⊗udν(u)≥expn Z

Sk−1

logt(u)dν(u)o

, (2.14)

with equality if and only if t(u1)· · ·t(uk) is constant for linearly independent unit vectors u1, . . . , uk suppν.

We shall need the following lemma due to Lutwak, Yang, and Zhang [22].

Lemma 2.4. If ν is an isotropic measure on Sk−1 in Rk and h∈L2(ν), then

°°

° Z

Sk−1

uh(u)dν(u)

°°

°

³ Z

Sk−1

h(u)2dν(u)

´1/2 . 3. Asymmetric cases

Theorem 1.1 immediately follows from Theorems 3.1 and 3.2.

Theorem 3.1. Let H be a k-dimensional subspace of Rn with 1≤k ≤n. Suppose thatf is a positive bounded measurable function on Sn−1 and that the measure ν on Sn−1 is isotropic andf-centered with respect to H. If the support subset ΨH of ν is measurable, then

|H∩Wν,f| ≤ (k+ 1disp(H∩Wν,f))k+1

k!(k+ 1)k+12 kfkkL2H), (3.1) with equality if and only ifH∩Wν,f is a regular simplex inscribed in kfkL2(ΨHk )(Sn−1 H) and kPf(u)

Huk is constant for all u∈ΨH.

Proof. By (1.1), (1.7) and (1.8), we may assume that kfkL2H) = 1.

Taking ϕ = (−w,f(w)) in (2.8), Lemma 2.2 yields that ˜¯ ν defined in (2.9) is isotropic on Sn∩H0.

By (2.5), we can consider ¯W¯ν,f¯ instead of H∩Wν,f. Define the cone C H0 = span{en+1} by

C = [

r>0

rW¯¯ν,f¯× {r} ⊂H0.

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Obviously, en+1 ∈C. By (2.9), η∈supp ˜ν if and only if η = (−w,f¯(w))

p1 + ¯f2(w) (3.2)

for some w supp ¯ν. Moreover, it follows from the definition of ¯Wν,¯f¯ that, for every η supp ˜ν and z = (y, r)∈C,

η·z = −w·y+rf¯(w) p1 + ¯f2(w) 0.

Now, for η∈supp ˜ν, define the strictly increasing functionφη : (0,∞)→R by Z φη(τ)

−∞

e−πs2ds= 1 en+1·η

Z τ

0

exp

³

s

en+1·η

´ ds.

Differentiating both sides with respect to τ, taking logarithms, and putting τ = η·z for η∈supp ˜ν and z int C, we get

logφ0η·z)−πφ2η·z) =−log(en+1·η)− η·z

en+1·η. (3.3) Define the transformation T : int C→H0 by

T(z) = Z

Sn∩H0

φη·z)ηdν(η).˜ Hence, for every z int C,

dT(z) = Z

Sn∩H0

φ0η·z)η⊗ηd˜ν(η). (3.4) Sinceφ0η >0, the matrixdT(z) is positive definite forz intC. Therefore,T : intC→ H0 is injective. Moreover, applying Lemma 2.4 with h(η) =φη·z) yields

kT(z)k2 Z

Sn∩H0

φη·z)2d˜ν(η). (3.5) From (3.3), the Ball-Barthe inequality (2.14) with t(η) = φ0η·z), (3.4), (3.5), and the change of variables x=T(z), we obtain

Z

int C

exp

³ Z

Sn∩H0

³

log(en+1·η)− η·z en+1·η

´ d˜ν(η)

´ dz

= Z

int C

exp³ Z

Sn∩H0

−πφ2η·z)d˜ν(η)´

exp³ Z

Sn∩H0

logφ0η·z)dν(η)˜ ´ dz

Z

int C

exp(−πkT(z)k2)detdT(z)dz Z

H0

e−πkxk2dx= 1. (3.6)

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On the other hand, the isotropy of ˜ν on Sn∩H0 yields ˜ν(Sn∩H0) = k+ 1. So applying Jensen’s inequality, we arrive at

exp

³ Z

Sn∩H0

log(en+1·η)d˜ν(η)

´

= exp

³ Z

Sn∩H0

1

k+ 1log(en+1·η)2d˜ν(η)

´k+1

2

exp h

log

³ Z

Sn∩H0

1

k+ 1(en+1·η)2d˜ν(η)

´ik+1

2

=

³ 1 k+ 1

´k+1

2 . (3.7)

By (2.9), (3.2), (2.10), and (2.11), we obtain that, for z = (y, r)∈C,

Z

Sn∩H0

(y, r)·η

en+1·ηd˜ν(η) = Z

Sn−1∩H

(y, r)·(−w,f(w))/¯ p

1 + ¯f2(w) en+1·(−w,f(w))/¯ p

1 + ¯f2(w)(1 + ¯f2(w))d¯ν(w)

= Z

Sn−1∩H

³

y·w

f(w)¯ +r−(y·w) ¯f(w) +rf¯2(w)

´ d¯ν(w)

= Z

Sn−1∩H

y·w

f¯(w)d¯ν(w) + (k+ 1)r. (3.8)

Applying Jensen’s inequality and (2.12), we get

1

|rW¯¯ν,f¯| Z

rW¯ν,¯f¯

exp

³ Z

Sn−1∩H

y·w f(w)¯ d¯ν(w)

´ dy

exp

³ 1

|rW¯ν,¯f¯| Z

rW¯ν,¯f¯

Z

Sn−1∩H

y·w

f¯(w)d¯ν(w)dy

´

= exp(rdisp( ¯W¯ν,f¯)). (3.9)

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Thus, by (3.7), (3.8), and (3.9), we have Z

int C

exp

³ Z

Sn∩H0

³

log(en+1·η)− η·z en+1·η

´ d˜ν(η)

´ dz

(k+ 1)k+12 Z

int C

exp

³ Z

Sn−1∩H

y·w

f(w)¯ d¯ν(w)−(k+ 1)r

´ dz

= (k+ 1)k+12 Z

0

e−(k+1)r Z

rW¯ν,¯f¯

exp

³ Z

Sn−1∩H

y·w f(w)¯ d¯ν(w)

´ dydr

(k+ 1)k+12 Z

0

e−(k+1)rexp

³

rdisp( ¯Wν,¯f¯)

´

|rW¯ν,¯f¯|dr.

= (k+ 1)k+12 |W¯ν,¯f¯| Z

0

e−(k+1−disp( ¯Wν,¯f¯))rrkdr.

= (k+ 1)k+12 |W¯ν,¯f¯|k!(k+ 1disp( ¯Wν,¯f¯))−(k+1).

Note that by (2.12), disp( ¯Wν,¯f¯) k. This, together with (3.6), (2.5), and (2.13), yields that

|H∩Wν,f| ≤ (k+ 1disp(H∩Wv,f))k+1 k!(k+ 1)k+12 , which is the desired inequality.

Assume that there is equality in inequality (3.1). By the equality conditions of Jensen’s inequality, equality in (3.7) holds if and only if en+1·η is constant for any η supp ˜ν. It follows from (3.2) that ¯f is constant on supp ¯ν, which, by (2.6), means that the function kPf(u)

Huk is constant for all u ΨH. Moreover, by (2.11), we must have ¯f(w) = 1kkfkL2H) on supp ¯ν.

Since ˜ν is isotropic on Sn ∩H0, there exist linearly independent η1, . . . , ηk+1 supp ˜ν such that

1, . . . , ηk+1} ⊆supp ˜ν.

Assume that there exists a different vector η0 supp ˜ν. Write η0 = λ1η1 +· · ·+ λk+1ηk+1 such that at least one coefficient, say λ1, is not zero. Then the equality conditions of the Ball-Barthe inequality imply that

φ0η1(z·η10η2(z·η2)· · ·φ0ηk+1(z·ηk+1) =φ0η0(z·η00η2(z·η2)· · ·φ0ηk+1(z·ηk+1), for all z ∈H0. But φ0η >0, and hence

φ0η1(z·η1) = φ0η0(z·η0)

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for all z ∈H0. Differentiating both sides with respect to z shows that φ00η1(z·η11 =φ00η0(z·η00,

for all z H0. Since there exists z H0 such that φ00η1(z · η1) 6= 0, it follows that η0 = ±η1. But ˜ν is supported inside a hemisphere of Sn ∩H0, so η0 = η1. Consequently,

1, . . . , ηk+1}= supp ˜ν.

Therefore, we have fory ∈H0

|y|2 = Xk+1

i=1

˜

ν({ηi})|y·ηi|2.

Substituting y =ηj Sn∩H0, we see that necessarily ˜ν({ηj}) 1. From the fact that Pk+1

i=1 ν({η˜ i}) =k+ 1 we get ˜ν({ηj}) = 1. Thus, ηj ·ηi = 0 forj 6=i. That is, η1, . . . , ηk+1 is an orthonormal basis of H0.

From (2.9) and (3.2), it follows that supp ¯ν ={w1, . . . , wk+1}, where ηi = (−wi,f(w¯ i))

p1 + ¯f2(wi), 1≤i≤k+ 1.

Moreover, we have

0 = ηi·ηj = (−wi,f¯(wi))

p1 + ¯f2(wi) · (−wj,f¯(wj))

p1 + ¯f2(wj), 1≤i6=j ≤k+ 1.

That is, wi ·wj = −f(w¯ i) ¯f(wj) for all i 6= j. Since ¯f(w) = kfkL2(ΨHk ), we obtain that wi ·wj = kfk

2 L2(ΨH)

k , i 6= j. By the normalization kfkL2H) = 1, we have wi·wj =k1 for alli6=j. Hence, conv(supp ¯ν) must be a regular simplex inscribed in the subsphere Sn−1 ∩H. Now (2.5) implies that H∩Wν,f is a regular simplex

inscribed in kfkL2(ΨHk )(Sn−1∩H). ¤

Theorem 3.2. Under the conditions of Theorem 3.1, we have

|PHWν,f | ≥ (k+ 1)k+12

k! kfk−kL2H),

with equality if and only ifH∩Wν,f is a regular simplex inscribed in kfkL2(ΨHk )(Sn−1 H) and kPf(u)

Huk is constant for all u∈ΨH.

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Proof. By (1.1) and (1.8), we may assume that kfkL2H)= 1.

Taking ϕ+ = (w,f(w)) in (2.8), Lemma 2.2 yields that ˜¯ ν defined in (2.9) is isotropic on Sn ∩H0. For η supp ˜ν, define the strictly increasing function φη : R(0,∞) by

en+1·η

Z φη(τ)

0

e−(en+1·η)sds = Z τ

−∞

e−πs2ds.

Differentiating both sides with respect to τ, taking logarithms, and putting τ = η·z for η∈supp ˜ν and z ∈H0, we get

log(en+1·η)−(en+1·η)φη·z) + logφ0η·z) = −π(η·z)2. (3.10) Define the transformation T :H0 →H0 by

T(z) = Z

Sn∩H0

φη·z)ηdν(η),˜ (3.11) for z ∈H0. The differential ofT is given by

dT(z) = Z

Sn∩H0

φ0η·z)η⊗ηd˜ν(η). (3.12) Since φ0η >0, the matrix dT(z) is positive definite for every z ∈H0. Therefore, T : H0 →H0 is injective.

Define the cone C⊂H0 =span{en+1} by C = [

r>0

rW¯¯ν,f¯× {r}.

Note that T(z) C for all z H0. To see this, it is sufficient to show that if T(z) = (y, r) ∈H0 and x∈ W¯ν,¯f¯, then y·x≤r. By (2.9) with ϕ+ given in(2.8), (3.11), and the fact that w·x≤f¯(w) for every w∈supp ¯ν, we have

y·x = Z

Sn−1∩H

φη

³ (w,f¯(w)) p1 + ¯f2(w)·z

´³ w

p1 + ¯f2(w) ·x

´

(1 + ¯f2(w))d¯ν(w)

Z

Sn−1∩H

φη

³ (w,f¯(w)) p1 + ¯f2(w)·z

´ f(w)¯

p1 + ¯f2(w)(1 + ¯f2(w))d¯ν(w)

= Z

Sn∩H0

φη·z)(η·en+1)d˜ν(η)

= T(z)·en+1 =r.

(14)

From (2.1), the fact that T(z) C for all z H0, the Ball-Barthe inequality (2.14) with t(η) =φ0η·z), (3.12), (3.11), (3.10), the isotropy of ˜ν on H0, Jensen’s inequality, and the isotropy of ˜ν again, we have

k!|W¯¯ν,f¯|= Z

H

e−kxkW¯ν,¯f¯dx= Z

0

Z

rW¯¯ν,f¯

e−rdxdr = Z

C

e−en+1·xdx

Z

H0

e−en+1·T(z)detdT(z)dz

Z

H0

exp

³

Z

Sn∩H0

φη·z)(en+1·η)d˜ν(η)

´ exp

³ Z

Sn∩H0

logφ0η·z)dν(η)˜

´ dz

= exp

³

Z

Sn∩H0

log(en+1·η)d˜ν(η)

´ Z

H0

exp

³

−π Z

Sn∩H0

·z)2d˜ν(η)

´ dz

= exp

³

Z

Sn∩H0

log(en+1·η)d˜ν(η)

´ Z

H0

e−πkzk2dz

= exp

³ Z

Sn∩H0

1

k+ 1log(en+1·η)2d˜ν(η)

´k+1

2

(k+ 1)k+12 .

Note that (see e.g., [12, (0.38)])

(H∩Wν,f) = PHWν,f , (3.13) where the polar operation on the left is taken in H. Thus, we have

|PHWν,f |=|(H∩Wν,f)|=|W¯ν,¯f¯| ≥ (k+ 1)k+12 k! .

The equality conditions are proved in basically the same way as in the proof of Theorem 3.1.

¤ 4. Symmetric cases

Theorem 1.2 immediately follows from the following two theorems.

Theorem 4.1. Let H be a k-dimensional subspace of Rn with 1≤k ≤n. Suppose thatf is an even positive bounded measurable function on Sn−1 and that the measure ν on Sn−1 is even isotropic. If the support subset ΨH of ν is measurable, then

|H∩Wν,f| ≤

³ 2

√k

´k

kfkkL2H),

(15)

with equality if and only if H∩Wν,f is an origin-symmetric cube with side length of

2

kkfkL2H) and kPf(u)

Huk is constant for all u∈ΨH.

Proof. Clearly,H∩Wν,f = ¯W¯ν,f¯is origin-symmetric and ¯ν and ¯f are both even.

For w supp ¯ν, define the strictly increasing function φw : (−f(w),¯ f(w))¯ R by

1 f¯(w)

Z τ

f(w)¯ 1[−f(w),¯ f(w)]¯ (s)ds= 1 Γ(32)

Z φw)

−∞

e−s2ds.

Differentiating both sides with respect to τ, taking logarithms on both sides and puttingτ =y·w for y∈W¯ν,¯f¯ and w∈supp ¯ν, we get

log Γ³3 2

´

log ¯f(w) + log1[−f¯(w),f(w)]¯ (y·w) =−φw(y·w)2+ logφ0w(y·w). (4.1) Define the transformation T : int ¯Wν,¯f¯→H by

T(y) = Z

Sn−1∩H

w(y·w)d¯ν(w). (4.2) Then, the differential of T is given by

dT(y) = Z

Sn−1∩H

w⊗wφ0w(y·w)d¯ν(w). (4.3) Sinceφ0w >0, the matrix dT(y) is positive definite for eachy∈int ¯Wν,¯f¯. Hence, the transformation T : int ¯W¯ν,f¯→H is injective.

Applying Lemma 2.4 with h(w) =φw(y·w) yields kT(y)k2

Z

Sn−1∩H

φw(y·w)2d¯ν(w). (4.4) The definition of ¯W¯ν,f¯implies that, for all y∈W¯¯ν,f¯,

exp

³ Z

Sn−1∩H

log1[−f¯(w),f(w)]¯ (y·w)d¯ν(w)

´

= 1. (4.5)

From (4.5), (4.1), (2.4), (4.4), the Ball-Barthe inequality (2.14) with t(w) = φ0w(w·y), (4.3), and the change of variablez =T y, we have

|W¯¯ν,f¯|= Z

int ¯Wν,¯f¯

dy

= Z

int ¯Wν,¯f¯

exp

³ Z

Sn−1∩H

log1[−f¯(w),f(w)]¯ (y·w)d¯ν(w)

´ dy

(16)

= Γ

³3 2

´k−n exp

³ Z

Sn−1∩H

log ¯f(w)d¯ν(w)

´

× Z

int ¯Wν,¯f¯

exp

³ Z

Sn−1∩H

−φw(y·w)2d¯ν(w)

´ exp

³ Z

Sn−1∩H

logφ0w(y·w)d¯ν(w)

´ dy

Γ

³3 2

´k−n exp

³ Z

Sn−1∩H

log ¯f(w)d¯ν(w)

´ Z

int ¯W¯ν,f¯

e−kT yk2detdT(y)dy

Γ

³3 2

´k−n exp

³ Z

Sn−1∩H

log ¯f(w)d¯ν(w)

´ Z

H

e−kzk2dz

= 2kexp

³ Z

Sn−1∩H

log ¯f(w)d¯ν(w)

´ .

Applying Jensen’s inequality and (2.11), we get exp

³ Z

Sn−1∩H

log ¯f(w)d¯ν(w)

´

= h

exp

³1 k

Z

Sn−1∩H

log ¯f(w)d¯ν(w)

´ik

³1 k

Z

Sn−1∩H

f¯2(w)d¯ν(w)

´k/2

= 1

kk/2kfk¯ kL2ν) = 1

kk/2kfkkL2H). (4.6) Therefore, we obtain the desired inequality

|H∩Wν,f|=|W¯¯ν,f¯| ≤

³ 2

√k

´k

kfkkL2H).

Assume that there is equality in inequality (3.1). As in the proof of Theorem 3.1, it is easy to verify that ¯ν is a cross measure on Sn−1 ∩H. By the equality conditions of Jensen’s inequality, equality in (4.6) holds if only if ¯f(w) is constant for everyw∈supp ¯ν. Hence, by (2.6), the function kPf(u)

Huk is constant for allu∈ΨH. Moreover, by (2.11), we must have ¯f(w) = 1kkfkL2H) on supp ¯ν. It follows from (2.5) thatH∩Wν,f is an origin-symmetric cube with side length of 2kkfkL2H). ¤ The following lemma was proved by Lutwak, Yang, and Zhang [21, Lemma 3.1].

Lemma 4.2. Let ν¯ be an even Borel measure on Sn−1∩H and let W¯ν,¯f¯be a Wulff shape on H. Ifh∈L1(ν), then

°°

° Z

Sn−1∩H

wh(w)dν(w)

°°

°W¯ν,¯f¯

Z

Sn−1∩H

f¯(w)|h(w)|dν(w), (4.7) where the polar operation on W¯ν,¯f¯is taken in H.

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