letBpn denote the unit ball of the`np-space, that is, Bnp = n x∈Rn: ³Xn i=1 |x·ei|p ´1 p ≤1 o , 1≤p &lt

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SECTIONS AND PROJECTIONS OF Lp-ZONOIDS AND THEIR POLARS

AI-JUN LI, QINGZHONG HUANG, AND DONGMENG XI

Abstract. Let 1≤ k≤ n. Sharp upper and lower bounds for the volume of k-dimensional projections (or sections) ofLp-zonoids (or their polars) with even isotropic generating measures are established. As special cases, sharp volume inequalities fork-dimensional sections and projections of`ⁿ_p-balls are recovered. The necessary conditions of equalities are new.

1. Introduction

We shall use | · | to denote k-dimensional volume (Lebesgue measure on the corresponding subspace), andk · kfor the standard Euclidean norm on R^k,k= 1, . . . , n. For 1≤p≤ ∞, letB_pⁿ denote the unit ball of the`ⁿ_p-space, that is,

Bⁿ_p = n

x∈Rⁿ:

³Xⁿ

i=1

|x·ei|^p

´¹

p ≤1 o

, 1≤p <∞,

B_∞ⁿ ={x∈Rⁿ:|x·ei| ≤1, i= 1, . . . , n}, p=∞,

where{e1, . . . , en}is the canonical basis of Rⁿ andx·ei denotes the standard inner product ofx andei.

Estimating the volumes of sections and projections of `ⁿ_p-balls has attracted much attention.

In [18], Hensley showed that ifH is an (n−1)-dimensional subspace ofRⁿ then 1≤ |H∩B_∞ⁿ|

|B∞ⁿ⁻¹| ≤5, and conjectured that the best upper bound is√

2. Ball [2] solved this conjecture and further proved that [3] ifHis ak-dimensional subspaces, then the upper bound is (√

2)^n−k, being best possible for someH ifn≥2(n−k). He also showed that ifkdividesn, the upper bound (ⁿ_k)^k² is best possible.

However, whenk < n/2 andk is not a divisor ofn, the best upper bound is still unknown. The lower bound fork-dimensional subspaces was obtained by Vaaler [44] who showed that it is always

2000Mathematics Subject Classification. 52A40.

Key words and phrases. Lp-zonoid, section, projection,`ⁿp-ball.

The first author was supported by NSFC-Henan Joint Fund (No. U1204102). The second author was supported in part by the National Natural Science Foundation of China (No. 11371239). The third author was supported by Shanghai Sailing Program (No. 16YF1403800).

1

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bigger than 1. His method was pushed further by Meyer and Pajor [37] who proved that for any k-dimensional subspaceH ofRⁿ, 1≤k≤n, and for every 1≤q≤p≤ ∞,

|H∩B_qⁿ|

|B_q^k| ≤ |H∩B_pⁿ|

|B^k_p| . (1)

The Meyer-Pajor inequality (1) immediately yields an upper bound for sections ofB_pⁿfor 1≤p≤2.

The case 2≤p≤ ∞was established by Barthe [6]:

|H∩B_pⁿ|

|B_p^k| ≤







1, 1≤p≤2,

³n k

´k(¹₂−¹_p)

, 2≤p≤ ∞.

(2)

Especially, ifk dividesn, the constant in (2) is best possible for 2≤p≤ ∞.

The dual version of the Meyer-Pajor inequality for (n−1)-dimensional subspaces was established by Barthe and Naor [7]: Let 1 ≤q ≤ p≤ ∞ and let PH be the orthogonal projection onto an (n−1)-dimensional subspaceH ofRⁿ. Then

|P_HB_qⁿ|

|Bⁿ⁻¹q | ≤ |P_HBⁿ_p|

|Bpⁿ⁻¹|. (3)

For the orthogonal projection PH onto ak-dimensional subspaceH ofRⁿ, Barthe [6] showed that

|PHB_pⁿ|

|B_p^k| ≥







³n k

´_k(¹

2−_p¹)

, 1≤p≤2, 1, 2≤p≤ ∞.

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Especially, ifkdividesn, the constant in (4) is best possible for 1≤p≤2. For more information about sections and projections of`ⁿ_p-balls, see, e.g., [8, 12, 13, 19, 36, 39, 40, 46].

The results described above motivated us to investigate the volumes of sections and projections of more general convex bodies than`ⁿ_p-balls. We will considerLp-zonoids and their polar, which were introduced by Schneider and Weil [42], and thoroughly investigated by Lutwak, Yang, and Zhang [32]. Note that`ⁿ_p-balls are special cases of polars ofLp-zonoids.

A Borel measure (always assumed to be nonnegative and finite)µonSⁿ⁻¹isisotropic if Z

Sⁿ⁻¹

u⊗udµ(u) =In, (5)

whereu⊗uis the rank-one orthogonal projection onto the space spanned byuandInis the identity map on Rⁿ. Note that it is impossible for an isotropic measure to be concentrated on a proper subspace ofRⁿ. The measureµis said to beeven if it assumes the same value on antipodal sets.

Across measure onSⁿ⁻¹, as a special case of even isotropic measures, is concentrated uniformly on{±u₁, . . . ,±u_n}, whereu₁, . . . , u_n is an orthonormal basis ofRⁿ.

Assume that µ is an even Borel measure onSⁿ⁻¹, whose support, suppµ, is not contained in a subsphere of Sⁿ⁻¹. For each 1 ≤p ≤ ∞, define the Lp-zonoid Zp = Zp(µ) (with generating

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p

measureµ) to be the origin-symmetric convex body in Rⁿ whose support function, forx∈Rⁿ, is given by

hZp(x) =³ Z

Sⁿ⁻¹

|x·u|^pdµ(u)´¹

p, 1≤p <∞, and forp=∞, is given by

hZ∞(x) = lim

p→∞hZp(x) = max

u∈suppµ|x·u|.

The bodiesZ1are usually called zonoids, and are precisely the class of bodies obtained as limits of Minkowski sums of line segments. Lp-zonoids belong to theLpBrunn-Minkowski theory, beginning largely with [26, 27], and expanding rapidly thereafter (see, e.g., [1, 10, 11, 15–17, 20, 23–35, 38, 45]).

A series of papers by Ball [4], Barthe [5], and Lutwak, Yang, and Zhang [32], provide sharp upper and lower estimates for the volumes ofLp-zonoidsZp and their polarsZ_p^∗, under the assumption that their generating measure is even and isotropic. In this paper we will establish the sharp upper and lower bounds for the volume ofk-dimensional projections (or sections) ofLp-zonoids (or their polars) with even isotropic generating measures.

Denote byωn the volume of the Euclidean unit ballB₂ⁿ in Rⁿ. For 1≤k≤n, definecp,k by cp,k=

ÃΓ(1 +^k₂)Γ(^1+p₂ ) Γ(1 +¹₂)Γ(^k+p₂ )

!^k

p

, 1≤p <∞,

and define c∞,k = limp→∞cp,k = 1. For 1 ≤ p≤ ∞, let p^∗ denote the H¨older conjugate of p;

i.e., 1/p+ 1/p^∗ = 1. If µ is a Borel measure on Sⁿ⁻¹, then the restriction of µ to a Borel set A⊆Sⁿ⁻¹is denoted by µxA. LetH^⊥ denote the orthogonal complement of a subspaceH ofRⁿ. The main results of this paper are the following. The case k=n was proved by Lutwak, Yang, and Zhang [32].

Theorem 1.1. Let 1 ≤ k ≤ n and let H be a k-dimensional subspace of Rⁿ. If µ is an even isotropic measure onSⁿ⁻¹, then

ωk

cp,k

³µ(Sⁿ⁻¹\H^⊥) k

´₋^k

p ≤ |H∩Z_p^∗(µ)| ≤







|B^k_p|, 1≤p≤2,

³n k

´_k(¹

2−_p¹)

|B^k_p|, 2≤p≤ ∞,

(6)

and

ωkcp,k

³µ(Sⁿ⁻¹\H^⊥) k

´^k

p ≥ |PHZp(µ)| ≥







|B_p^k∗|, 1≤p≤2,

³n k

´k(¹₂−_p¹_∗)

|B_p^k∗|, 2≤p≤ ∞.

(7) For 1 ≤ p < 2, there is equality in the right side inequalities of (6) or (7) if and only if µ is concentrated on H∪H^⊥ and µx(Sⁿ⁻¹∩H) is a cross measure on Sⁿ⁻¹∩H. For 2 < p ≤ ∞, there is equality in the right side inequalities of (6) or (7) if and only if there is an orthonormal basis{ω1, . . . , ωk} of H such that

µ¡

Sⁿ⁻¹∩(H^⊥+p

k/nωi)¢

=µ¡

Sⁿ⁻¹∩(H^⊥−p

k/nωi)¢

= n

2k, i= 1, . . . , k. (8)

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There is equality in the left side inequalities of (6)or (7)if and only ifµis concentrated onH∪H^⊥ withµ(Sⁿ⁻¹\H^⊥) =k andH∩Z_p^∗ is a Euclidean ball inH.

The extremal measures for the inequalities of Theorem 1.1 have intuitive geometric meanings.

It is easy to see that, for 1≤ p <2, if µx(Sⁿ⁻¹∩H) is a cross measure on Sⁿ⁻¹∩H and µ is concentrated onH ∪H^⊥, thenH ∩Z_p^∗ is the ballB^k_p in H up to a rotation. When 2 < p≤ ∞, condition (8) tells us that the extremal measureµis concentrated on the 2kintersections of unit spheres and translations ofH^⊥. This also implies thatH∩Z_p^∗is a dilatation ofB^k_p up to a rotation (see Section 3 for details). Moreover, some special convex bodies are related to the extremals of the left sides of (6) and (7). For example, when k =n−1, the double cone for p= 1, and the cylinder forp=∞.

Our proof of Theorem 1.1, contained in Section 3, is based on a refinement of the approach by Lutwak, Yang, and Zhang [32], which, in turn, uses ideas of Ball [4] and Barthe [5]. More precisely, we apply the Ball-Barthe inequality and mass transportation. The approach we choose is self contained and elementary. For more applications about this approach, see e.g., [20–22, 32, 34, 43].

As special cases of Theorem 1.1, in Section 4, we will recover inequalities (2) and (4) in Theorems 4.1 and 4.2. We will show that the corresponding constants in (2) and (4) are best possibleonly if kdividesn.

2. Background materials

We list basic facts about convex bodies. As a general reference, we refer the reader to the books of Gardner [14] and Schneider [41].

Throughout this paper a convex bodyKin Euclidean n-spaceRⁿ is a compact convex set that contains the origin in its interior. Its polar bodyK^∗ is defined by

K^∗={x∈Rⁿ:x·y≤1 for ally∈K}.

The support function ofK,hK(·) :Rⁿ→(0,∞), is defined forx∈Rⁿ by hK(x) = max{x·y:y∈K}.

The Minkowski functionalk · kK of a convex body Kis defined by

kxkK = min{t >0 :x∈tK} (9)

and related to the support function ofKby

kxkK=hK^∗(x). (10)

For eachp∈(0,∞), the volume ofK is given by

|K|= 1 Γ(1 +ⁿ_p)

Z

Rⁿ

e^−kxk^p^Kdx, (11)

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p

where the integral is with respect to Lebesgue measure onRⁿ. In particular, we have

|B_pⁿ|=(2Γ(1 +¹_p))ⁿ Γ(1 +ⁿ_p) .

The polar coordinate formula for the volume ofK will also be used later:

|K|= 1 n

Z

Sⁿ⁻¹

kuk⁻ⁿ_K du, (12)

whereduis the spherical Lebesgue measure onSⁿ⁻¹.

Suppose that 1≤k≤n andH is a k-dimensional subspace ofRⁿ. Let µbe a Borel measure onSⁿ⁻¹. If ¯µis the Borel measure onSⁿ⁻¹∩H defined by

¯ µ(A) =

Z

Sⁿ⁻¹\H^⊥

1_A³ PHu kPHuk

´

kP_Huk²dµ(u) (13)

for Borel setsA⊂Sⁿ⁻¹∩H, then for any continuous functionf :Sⁿ⁻¹∩H→R, Z

Sⁿ⁻¹∩H

f(w)d¯µ(w) = Z

Sⁿ⁻¹\H^⊥

f³ PHu kPHuk

´

kPHuk²dµ(u). (14) Ifµis an isotropic measure onSⁿ⁻¹, then for an arbitraryy∈H, we have

Z

Sⁿ⁻¹∩H

(y·w)²d¯µ(w) = Z

Sⁿ⁻¹\H^⊥

³

y· PHu kPHuk

´₂

kPHuk²dµ(u)

= Z

Sⁿ⁻¹

(y·u)²dµ(u) =kyk². Hence, ¯µis isotropic onSⁿ⁻¹∩H. Moreover, we have

¯

µ(Sⁿ⁻¹∩H) =k. (15)

From the definition ofZpand (10), it is easy to see that for 1≤p <∞, H∩Z_p^∗=n

y∈H :³ Z

Sⁿ⁻¹\H^⊥

|y·PHu|^pdµ(u)´¹

p ≤1o

, (16)

and forp=∞,

H∩Z_∞^∗ ={y∈H: sup

u∈suppµ\H^⊥

|y·PHu| ≤1}. (17) Hence, for 1≤p <∞,

kykH∩Z^∗_p=h_(H∩Z_p^∗₎^∗(y) =

³ Z

Sⁿ⁻¹\H^⊥

|y·PHu|^pdµ(u)

´¹

p, y∈H. (18)

and forp=∞,

kyk_H∩Z_∞^∗ =h_(H∩Z∗

∞)^∗(y) = sup

u∈suppµ\H^⊥

|y·P_Hu|, y∈H. (19) LetLp(µ) denote the usualLpspace with respect to µ. Forf ∈L1(µ) definef^o∈H by

f^o= Z

Sⁿ⁻¹\H^⊥

PHukPHukf

³ PHu kPHuk

´

dµ(u) = Z

Sⁿ⁻¹∩H

wf(w)d¯µ(w).

The following lemma fork=nwas proved by Lutwak, Yang, and Zhang [32, Lemma 3.1].

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Lemma 2.1. Suppose that1≤k≤n. LetH be a k-dimensional subspace of Rⁿ and let µbe an even Borel measure onSⁿ⁻¹. Iff ∈Lp^∗(µ), then

kf^ok(H∩Z^∗_p)^∗ ≤

³ Z

Sⁿ⁻¹\H^⊥

¯¯

¯f

³ PHu kPHuk

´¯¯

¯^p

∗

kPHuk^p^∗dµ(u)

´¹

p∗

. (20)

Proof. For ally∈H, by the H¨older inequality we have

|y·f^o|=

¯¯

¯ Z

Sⁿ⁻¹\H^⊥

(y·PHu)kPHukf³ PHu kPHuk

´ dµ(u)

¯¯

¯

≤³ Z

Sⁿ⁻¹\H^⊥

kPHuk^p^∗

¯¯

¯f³ PHu kPHuk

´¯¯

¯^p

∗

dµ(u)´¹

p∗³ Z

Sⁿ⁻¹\H^⊥

|y·PHu|^pdµ(u)´¹

p

=

³ Z

Sⁿ⁻¹\H^⊥

kPHuk^p^∗

¯¯

¯f

³ PHu kPHuk

´¯¯

¯^p

∗

dµ(u)

´¹

p∗

h_(H∩Z^∗_p₎^∗(y).

This allows to conclude (20) because

kf^ok(H∩Z_p^∗)^∗ = sup{|y·f^o|:h(H∩Z_p^∗)^∗(y)≤1, y∈H}.

¤

Lemma 2.1 directly implies the following by settingp= 2 and taking (14) into account.

Lemma 2.2 ( [34]). Ifµ¯ is an isotropic measure onSⁿ⁻¹∩H andf ∈L2(¯µ), then

°°

° Z

Sⁿ⁻¹∩H

ωf(w)d¯µ(ω)

°°

°²≤ Z

Sⁿ⁻¹∩H

f(w)²d¯µ(w).

The following continuous version (including the equality conditions) of the Ball-Barthe inequality is due to Lutwak, Yang, and Zhang [32], extending the discrete case due to Ball and Barthe [5, Proposition 9].

Lemma 2.3. Let 1 ≤k ≤n and let H be a k-dimensional subspace of Rⁿ. If µ¯ is an isotropic measure onSⁿ⁻¹∩H, then for each positive continuous functionf onsupp ¯µ

expn Z

Sⁿ⁻¹∩H

logf(w)d¯µ(w)o

≤det Z

Sⁿ⁻¹∩H

f(w)w⊗wd¯µ(w), (21) with equality if and only iff(w1)· · ·f(wk)is constant for linearly independent unit vectorsw1, . . . , wk∈ supp ¯µ.

3. Proofs of the main results

|H∩Z_p^∗(µ)| ≤







|B_p^k|, 1≤p≤2,

³n k

´_k(¹

2−¹_p)

|B_p^k|, 2≤p≤ ∞.

(22)

(7)

p

For1≤p <2, there is equality if and only ifµis concentrated onH∪H^⊥ andµx(Sⁿ⁻¹∩H)is a cross measure onSⁿ⁻¹∩H. For2< p≤ ∞, there is equality if and only if there is an orthonormal basis{ω1, . . . , ωk} of H such that

µ¡

Sⁿ⁻¹∩(H^⊥+p

k/nωi)¢

=µ¡

Sⁿ⁻¹∩(H^⊥−p

k/nωi)¢

= n

2k, i= 1, . . . , k.

Proof. Case 1≤p <∞: Foru∈suppµ\H^⊥ and t∈R, define the strictly increasing function φ:R→Rby

kPHuk¹⁻²^p Γ(1 +¹_p)

Z _t

−∞

e⁻

¯¯_kP_H_uk¹⁻²ps

¯¯^p

ds= 1 Γ(³₂)

Z _φ(t)

−∞

e^−|s|²ds.

Differentiating both sides with respect totand taking logarithms gives log Γ³3

2

´

+ (1−2

p) logkPHuk −log Γ(1 +1

p)− kPHuk^p−2|t|^p=−φ(t)²+ logφ⁰(t). (23) DefineT :H →H by

T y= Z

Sⁿ⁻¹∩H

wφ(y·w)d¯µ(w) (24)

for eachy∈H, where the measure ¯µis defined by (13). Thus, the differential ofT is given by dT(y) =

Z

Sⁿ⁻¹∩H

w⊗wφ⁰(y·w)d¯µ(w). (25) Sinceφ⁰ >0, the matrixdT(y) is positive definite for eachy ∈H. Hence, the transformation T: H →H is injective. Moreover, from Lemma 2.2 withf(w) =φ(y·w) and (24), we have

kT yk²≤ Z

Sⁿ⁻¹∩H

φ(y·ω)²d¯µ(ω). (26)

The Ball-Barthe inequality (21) withf(w) =φ⁰(y·w) and (25) imply that exp

n Z

Sⁿ⁻¹∩H

logφ⁰(y·ω)d¯µ(ω) o

≤det(dT(y)). (27)

From (11), (18), (23) witht =y· _kP^P^H^u

Huk, (15), (14), (26), and (27), we obtain by making the change of variablez=T y, that

Γ

³ 1 + k

p

´

|H∩Z_p^∗|= Z

H

e^−kyk

p H∩Z∗

pdy

= Z

H

expn

− Z

Sⁿ⁻¹\H^⊥

|y·PHu|^pdµ(u)o dy

= Z

H

expn

− Z

Sⁿ⁻¹\H^⊥

h φ³

y· PHu kPHuk

´2

−logφ⁰³

y· PHu kPHuk

´

+ log Γ³3 2

´

+ (1−2

p) logkPHuk −log Γ(1 + 1 p)i

kPHuk²dµ(u)o dy

=

³Γ(1 +_p¹) Γ¡₃

2

¢ ´_k exp

n Z

Sⁿ⁻¹\H^⊥

h (2

p−1) logkPHuk i

kPHuk²dµ(u) o

× Z

H

expn Z

Sⁿ⁻¹∩H

−φ(y·ω)²d¯µ(ω)o expn Z

Sⁿ⁻¹∩H

logφ⁰(y·ω)d¯µ(ω)o dy

(8)

≤

³Γ(1 +_p¹) Γ¡₃

2

¢ ´_k³ exp

Z

Sⁿ⁻¹\H^⊥

(logkPHuk)kPHuk²dµ(u)

´²

p−1Z

H

e^{−kT yk}²det(dT(y))dy

≤³Γ(1 +_p¹) Γ¡₃

2

¢ ´_k³ exp

Z

Sⁿ⁻¹\H^⊥

(logkPHuk)kPHuk²dµ(u)´²

p−1Z

H

e^−kzk²dz

=³

2Γ(1 +1 p)´_k³

exp Z

Sⁿ⁻¹\H^⊥

p−1

.

Thus,

|H∩Z_p^∗|

|B_p^k| = Γ(1 +^k_p)|H∩Z_p^∗| (2Γ(1 +¹_p))^k ≤³

exp Z

Sⁿ⁻¹\H^⊥

p−1

. (28) An application of Jensen’s inequality yields

exp

³

− Z

Sⁿ⁻¹\H^⊥

(logkPHuk)kPHuk²dµ(u)

´

= h

exp

³1 k

Z

Sⁿ⁻¹\H^⊥

(logkPHuk⁻²)kPHuk²dµ(u)

´i^k

2

≤

³1 k

Z

Sⁿ⁻¹\H^⊥

dµ(u)

´^k

2 ≤

³n k

´^k

2, (29)

which together with the fact thatkPHuk ≤1 gives

³k n

´^k

2 ≤exp³ Z

Sⁿ⁻¹\H^⊥

(logkPHuk)kPHuk²dµ(u)´

≤1. (30)

If 1≤p≤2, then from (28) and the right hand side of inequality (30), we deduce

|H∩Z_p^∗|

|B^k_p| ≤1.

If 2≤p <∞, then from (28) and the left hand side of inequality (30), we obtain

|H∩Z_p^∗|

|B_p^k| ≤

³n k

´_k(¹

2−¹_p)

.

Now we prove the equality conditions. Assume that equality in (28) holds . Since ¯µis isotropic on Sⁿ⁻¹∩H, the measure ¯µis not concentrated on any subsphere ofSⁿ⁻¹∩H. Thus there exist linearly independent w1, . . . , wk ∈ supp ¯µ. Since µ is even, so is ¯µ. Hence, we have {±w1, . . . ,±wk} ⊆ supp ¯µ. Assume that there exists a vector v ∈ supp ¯µ such that v /∈ {±w1, . . . ,±wk}. Let v = λ1w1+· · ·+λkwk where at least one coefficient, sayλ1, is not zero. Then the equality conditions of the Ball-Barthe inequality imply that

φ⁰(y·w1)φ⁰(y·w2)· · ·φ⁰(y·wk) =φ⁰(y·v)φ⁰(y·w2)· · ·φ⁰(y·wk)

for ally∈H. Butφ⁰ >0, and henceφ⁰(y·w1) =φ⁰(y·v) for ally∈H. Ifp6= 2, then the function φ⁰is not constant. Differentiating both sides with respect toyshows thatφ⁰⁰(y·w1)w1=φ⁰⁰(y·v)v

(9)

p

for ally ∈H. Since there existsy∈H such thatφ⁰⁰(y·w1)6= 0 it follows thatv =±w1. Hence {±w1, . . . ,±wk}= supp ¯µ.Therefore, we have fory∈H

|y|²= Xk

i=1

¯

µ({±wi})(y·wi)².

Substituting y = wj ∈ Sⁿ⁻¹∩H, we see that necessarily ¯µ({±wj}) ≤ 1. From the fact that P_k

i=1µ({±w¯ i}) =kwe get ¯µ({±wj}) = 1. Thus,wj·wi= 0 forj 6=i, and we obtain that ¯µis a cross measure onSⁿ⁻¹∩H.

When 1≤p <2, by the right hand inequality of (30), we havekP_Huk= 1 foru∈suppµ\H^⊥. This means thatµconcentrates onH∪H^⊥.Hence, it follows from (13) that suppµ∩H = supp ¯µ and thusµx(Sⁿ⁻¹∩H) is a cross measure onSⁿ⁻¹∩H.

Conversely, suppose that µis concentrated on H∪H^⊥ andµx(Sⁿ⁻¹∩H) is a cross measure on Sⁿ⁻¹∩H. Without loss of generality, we assume that suppµ∩H = {±ω1, ...,±ωk}, where {ω1, ..., ωk}is an orthonormal basis. It follows from (16) that

H∩Z_p^∗= n

y∈H :

³ Z

Sⁿ⁻¹\H^⊥

|y·PHu|^pdµ(u)

´¹

p ≤1 o

= n

y∈H :

³X^k

i=1

|y·ωi|^p

´¹

p ≤1 o

. This means thatH∩Z_p^∗ is isometric toB_p^k,and hence equality in (22) holds for 1≤p <2.

When 2< p <∞, by the equality conditions of Jensen’s inequality, equalities in (29) imply that µ(Sⁿ⁻¹∩H^⊥) = 0 andkPHuk is a constant for eachu∈suppµ. From

Z

Sⁿ⁻¹\H^⊥

kPHuk²dµ(u) =nkPHuk²=k, we get kPHuk = p

k/n. Thus, µ is concentrated on {u ∈ Sⁿ⁻¹ : kPHuk = p

k/n}. We have proved that ¯µis a cross measure onSⁿ⁻¹∩H, so set supp ¯µ={±ω1, ...,±ωk},where{ω1, . . . , ωk} is an orthonormal basis onSⁿ⁻¹∩H. Since ¯µis even and isotropic, this implies that ¯µ({ω_i}) =

¯

µ({−ωi}) =¹₂.For anyA⊂Sⁿ⁻¹∩H,we have Z

Sⁿ⁻¹∩{kPHuk=√

k/nk}

1A

³ PHu kPHuk

´

kPHuk²dµ(u) = Z

Sⁿ⁻¹∩H

1A(ω)d¯µ(ω).

LettingA={±ωi}, i∈ {1, . . . , k}, and using thatµis even, we get µ¡

Sⁿ⁻¹∩(H^⊥+p

k/nωi)¢

=µ¡

Sⁿ⁻¹∩(H^⊥−p

k/nωi)¢

= n 2k.

Conversely, suppose that there is an orthonormal basis{ω1, . . . , ωk} of H such thatµ¡ Sⁿ⁻¹∩ (H^⊥+p

k/nωi)¢

=µ¡

Sⁿ⁻¹∩(H^⊥−p

k/nωi)¢

=_2kⁿ, i= 1, . . . , k.Thus, H∩Z_p^∗=

n y∈H :

³ Z

Sⁿ⁻¹

|y·PHu|^pdµ(u)

´¹

p ≤1 o

= n

y∈H :

³X^k

i=1

n k

¯¯

¯y· rk

nωi

¯¯

¯^p

´¹

p ≤1 o

(10)

=

³n k

´¹

2−¹_pn y∈H :

³X^k

i=1

|y·ωi|^p

´¹

p ≤1 o

. It follows that equality in (22) holds for 2< p <∞.

Casep=∞: Foru∈suppµ\H^⊥, define the strictly increasing functionφ:¡

−_kP¹

Huk,_kP¹

Huk

¢→

Rby

kPHuk Z _t

−_kP¹

H uk

1_[− ¹

kPH uk,_kP¹

H uk](s)ds= 1 Γ(³₂)

Z _φ(t)

−∞

e^−s²ds.

Differentiating both sides with respect totand taking logarithms gives log Γ³3

2

´

+ logkP_Huk+ log1_[− 1 kPH uk,_kP¹

H uk](t) =−φ(t)²+ logφ⁰(t). (31) By (17) we have

H∩Z_∞^∗ =n

y∈H : sup

u∈suppµ\H^⊥

¯¯

¯y· PHu kPHuk

¯¯

¯≤ 1 kPHuk

o .

Thus, for eachy∈H∩Z_∞^∗ , expn Z

Sⁿ⁻¹\H^⊥

log1_[− ¹

kPH uk,_kP¹

H uk]

³

y· PHu kPHuk

´

kPHuk²dµ(u)o

= 1. (32)

DefineT : relint(H∩Z_∞^∗)→H as in (24), where relint(H∩Z_∞^∗ ) is the relative interior ofH∩Z_∞^∗ . Note that for ally∈relint(H∩Z_∞^∗ ) and allw∈supp ¯µ, y·wis in the domain ofφ. Clearly, the matrixdT(y) in (25) is positive definite for eachy ∈relint(H∩Z_∞^∗ ) and the transformation T: relint(H∩Z_∞^∗)→H is injective.

From (32), (31) with t =y· _kP^P^H^u

Huk, (15), (26), and (27), we obtain by making the change of variablez=T yand (29), that

|H∩Z_∞^∗ |= Z

relint(H∩Z_∞^∗)

exp n Z

Sⁿ⁻¹\H^⊥

log1_[− ¹

kPH uk,_kP¹

H uk]

³

y· PHu kPHuk

´

kPHuk²dµ(u) o

dy

= Γ

³3 2

´_−k exp

n Z

Sⁿ⁻¹\H^⊥

−(logkPHuk)kPHuk²dµ(u) o

× Z

relint(H∩Z_∞^∗)

exp n Z

Sⁿ⁻¹∩H

−φ(y·ω)²d¯µ(u) o

exp n Z

Sⁿ⁻¹∩H

logφ⁰(y·w)d¯µ(w) o

dy

≤Γ³3 2

´_−k expn Z

Sⁿ⁻¹\H^⊥

−(logkPHuk)kPHuk²dµ(u)o Z

relint(H∩Z^∗_∞)

e^{−kT yk}²det(dT(y))dy

≤Γ

³3 2

´_−k exp

n Z

Sⁿ⁻¹\H^⊥

−(logkPHuk)kPHuk²dµ(u) o Z

H

e^−kzk²dz

≤2^k³n k

´^k

2.

Therefore, we obtain the desired inequality

|H∩Z_∞^∗|

|B_∞^k | ≤³n k

´^k

2.

The proof of the equality conditions is almost the same as in the case 2< p <∞. ¤

(11)

p

|P_HZ_p(µ)| ≥







³n k

´_k(¹

2−_p¹∗)

|B_p^k∗|, 1≤p^∗≤2,

|B_p^k∗|, 2≤p^∗≤ ∞.

(33)

For 1 ≤p^∗ <2, there is equality if and only if there is an orthonormal basis {ω1, . . . , ωk} of H such that

µ¡

Sⁿ⁻¹∩(H^⊥+p

k/nωi)¢

=µ¡

Sⁿ⁻¹∩(H^⊥−p

k/nωi)¢

= n

2k, i= 1, . . . , k.

For2< p^∗ ≤ ∞, there is equality if and only if µis concentrated on H∪H^⊥ and µx(Sⁿ⁻¹∩H) is a cross measure onSⁿ⁻¹∩H.

Proof. Case 1≤p^∗ <∞: Foru∈suppµ\H^⊥ andt∈R, define the strictly increasing function φ:R→Rby

1 Γ(³₂)

Z _t

−∞

e^−|s|²ds=kPHuk¹⁻^p²^∗ Γ(1 +_p¹∗)

Z _φ(t)

−∞

e⁻

¯¯_kP_H_uk¹⁻p²∗s

¯¯^p^∗ ds.

Differentiating both sides with respect totand taking logarithms gives

−t²= log Γ

³3 2

´

+ (1− 2

p^∗) logkPHuk −log Γ(1 + 1

p^∗)− kPHuk^p^∗⁻²|φ(t)|^p^∗+ logφ⁰(t). (34) DefineT :H →H by

T y= Z

Sⁿ⁻¹∩H

wφ(y·w)d¯µ(w), (35)

for eachy∈H, where the measure ¯µis defined by (13). Thus, the differential ofT is given by dT(y) =

Z

Sⁿ⁻¹∩H

w⊗wφ⁰(y·w)d¯µ(w). (36) Sinceφ⁰ >0, the matrixdT(y) is positive definite for eachy ∈H. Hence, the transformation T: H →H is injective. Moreover, from Lemma 2.1 withf(w) =φ(y·w) and (35), we have

kT yk^p_(H∩Z^∗ ∗ p)^∗ ≤

Z

Sⁿ⁻¹\H^⊥

¯¯

¯φ³

y· PHu kPHuk

´¯¯

¯^p

∗

kPHuk^p^∗dµ(u). (37) The Ball-Barthe inequality (21) withf(w) =φ⁰(y·w) and (36) imply that

exp n Z

Sⁿ⁻¹∩H

logφ⁰(y·ω)d¯µ(ω) o

≤det(dT(y)). (38)

From (11), (34) with t = y · _kP^P^H^u

Huk, (15), (37), (38), and by making the change of variable z=T y, using (11) again, we have

Γ

³1 2

´_k

= Z

H

e^−kyk²dy= Z

H

exp n

− Z

Sⁿ⁻¹\H^⊥

³

y· PHu kPHuk

´₂

kPHuk²dµ(u) o

dy

= Z

H

exp n

− Z

Sⁿ⁻¹\H^⊥

h

kPHuk^p^∗⁻²

¯¯

¯φ

³

y· PHu kPHuk

´¯¯

¯^p

∗

−logφ⁰

³

y· PHu kPHuk

´

−log Γ

³3 2

´