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Geom Dedicata (2009) 141:109–122 DOI 10.1007/s10711-008-9346-x ORIGINAL PAPER On intersection and mixed intersection bodies Chang-jian Zhao Received: 24 March 2008 / Accepted: 28 November 2008 / Published online: 27 December 2008 © Springer Science+Business Media B.V. 2008 Abstract Duals of the basic projection and mixed projection inequalities are established for intersection and mixed intersection bodies. Keywords Dual mixed volumes · Intersection bodies · Mixed intersection bodies Mathematics Subject Classification (2000) 52A40 · 53A15 The intersection operator and the class of intersection bodies were defined by Lutwak [23]. The closure of the class of intersection bodies was studied by Goodey et al. [13]. The intersection operator and the class of intersection bodies played a critical role in Gardner [7] and Zhang [30] solution of the famous Busemann–Petty problem in three dimensions and four dimensions, respectively. (See also Gardner etal. [11].) Koldobsky’s book [19] where the history is described in detail about the solution of the famous Busemann–Petty problem. Just as the period from the mid-1960s to the mid-1980s was a time of great advances in the understanding of the projection operator and the class of projection bodies, during the past 15 years significant advances have been made in our understanding of the intersection operator and the class of intersection bodies by Koldobsky, Zhang, Campi, Goodey, Gardner, Grinberg, Fallert, Weil, and others (see, e.g., [26, 810, 1219, 2536]). As Lutwak [23] shows (and as is further elaborated in Gardner’s book [8]), there is a duality between projection and intersection bodies (that at present is not yet understood). Consider the following illustrative example: It is well known that the projections (onto lower dimen- sional subspaces) of projection bodies are themselves projection bodies. Lutwak conjectured the “dual”: When intersection bodies are intersected with lower dimensional subspaces, the results are intersection bodies (within the lower dimensional subspaces). This was proven by Fallert et al. [4]. In this paper new contributions that illustrate this mysterious duality will be presented. C.-j. Zhao (B ) Department of Information and Mathematics Sciences, College of Science, China Jiliang University, Hangzhou 310018, People’s Republic of China e-mail: [email protected]; [email protected]; [email protected] 123

On intersection and mixed intersection bodies

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Page 1: On intersection and mixed intersection bodies

Geom Dedicata (2009) 141:109–122DOI 10.1007/s10711-008-9346-x

ORIGINAL PAPER

On intersection and mixed intersection bodies

Chang-jian Zhao

Received: 24 March 2008 / Accepted: 28 November 2008 / Published online: 27 December 2008© Springer Science+Business Media B.V. 2008

Abstract Duals of the basic projection and mixed projection inequalities are establishedfor intersection and mixed intersection bodies.

Keywords Dual mixed volumes · Intersection bodies · Mixed intersection bodies

Mathematics Subject Classification (2000) 52A40 · 53A15

The intersection operator and the class of intersection bodies were defined by Lutwak[23]. The closure of the class of intersection bodies was studied by Goodey et al. [13]. Theintersection operator and the class of intersection bodies played a critical role in Gardner [7]and Zhang [30] solution of the famous Busemann–Petty problem in three dimensions andfour dimensions, respectively. (See also Gardner et al. [11].) Koldobsky’s book [19] wherethe history is described in detail about the solution of the famous Busemann–Petty problem.

Just as the period from the mid-1960s to the mid-1980s was a time of great advances inthe understanding of the projection operator and the class of projection bodies, during thepast 15 years significant advances have been made in our understanding of the intersectionoperator and the class of intersection bodies by Koldobsky, Zhang, Campi, Goodey, Gardner,Grinberg, Fallert, Weil, and others (see, e.g., [2–6,8–10,12–19,25–36]).

As Lutwak [23] shows (and as is further elaborated in Gardner’s book [8]), there is a dualitybetween projection and intersection bodies (that at present is not yet understood). Considerthe following illustrative example: It is well known that the projections (onto lower dimen-sional subspaces) of projection bodies are themselves projection bodies. Lutwak conjecturedthe “dual”: When intersection bodies are intersected with lower dimensional subspaces, theresults are intersection bodies (within the lower dimensional subspaces). This was proven byFallert et al. [4]. In this paper new contributions that illustrate this mysterious duality will bepresented.

C.-j. Zhao (B)Department of Information and Mathematics Sciences, College of Science, China Jiliang University,Hangzhou 310018, People’s Republic of Chinae-mail: [email protected]; [email protected]; [email protected]

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In [21] (see also [22] and [24]), Lutwak introduced mixed projection bodies and derivedtheir fundamental inequalities. In this work we shall derive, for intersection bodies, all theanalogous inequalities for Lutwak’s mixed projection body inequalities. Thus, this work maybe seen as presenting additional evidence of the duality between intersection and projectionbodies.

1 Notation and preliminaries

The setting for this paper is n-dimensional Euclidean space Rn(n > 2). Let C

n denote theset of non-empty convex figures (compact, convex subsets) and Kn denote the subset ofC

n consisting of all convex bodies (compact, convex subsets with non-empty interiors) inR

n . We reserve the letter u for unit vectors, and the letter B is reserved for the unit ballcentered at the origin. The surface of B is Sn−1. For u ∈ Sn−1, let Eu denote the hyper-plane, through the origin, that is orthogonal to u. We will use K u to denote the image ofK under an orthogonal projection onto the hyperplane Eu . We use V (K ) for the n-dimen-sional volume of convex body K . The support function of K ∈ Kn , h(K , ·), defined on R

n byh(K , ·) = Max{x · y : y ∈ K }. Let δ denote the Hausdorff metric on Kn ; i.e., for K , L ∈ Kn,

δ(K , L) = |hK − hL |∞,

where | · |∞ denotes the sup-norm on the space of continuous functions, C(Sn−1).

Associated with a compact subset K of Rn , which is star-shaped with respect to the origin,

is its radial function ρ(K , ·) : Sn−1 → R, defined for u ∈ Sn−1, by ρ(K , u) = Max{λ ≥0 : λu ∈ K }. If ρ(K , ·) is positive and continuous, K will be called a star body. Let ϕn

denote the set of star bodies in Rn . Let δ denote the radial Hausdorff metric, as follows, if

K , L ∈ ϕn , then

δ(K , L) = |ρK − ρL |∞.

1.1 Dual mixed volumes

Now introduce a vector addition on Rn , which we call radial addition, as follows. If

x1, . . . , xr ∈ Rn , then x1+ · · · +xr is defined to be the usual vector sum of x1, . . . , xr ,

provided x1, . . . , xr all lie in one-dimensional subspace of Rn , and as the zero vector other-

wise.If K1, . . . , Kr ∈ ϕn and λ1, . . . , λr ∈ R, then the radial Minkowski linear combination,

λ1 K1+ · · · + λr Kr , is defined by λ1 K1+ · · · + λr Kr = {λ1x1+ · · · + λr xr : xi ∈ Ki }. It hasthe following important property, for K , L ∈ ϕn and λ,µ ≥ 0

ρ(λK +µL , ·) = λρ(K , ·) + µρ(L , ·) (1.1.1)

For K1, . . . , Kr ∈ ϕn and λ1, . . . , λr ≥ 0, the volume of the radial Minkowski linercombination λ1 K1+ · · · + λr Kr is a homogeneous nth-degree polynomial in the λi ,

V (λ1 K1+ · · · + λr Kr ) =∑

Vi1,...,in λi1 . . . λin (1.1.2)

where the sum is taken over all n-tuples (i1, . . . , in) whose entries are positive integers notexceeding r . If we require the coefficients of the polynomial in (1.1.2) to be symmetric intheir arguments, then they are uniquely determined. The coefficient Vi1,...,in is nonnegativeand depends only on the bodies Ki1 , . . . , Kin . It is written as V (Ki1 , · · · , Kin ) and is calledthe dual mixed volume of Ki1 , . . . , Kin . If K1 = · · · = Kn−i = K , Kn−i+1 = · · · = Kn =

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L , the dual mixed volumes is written as Vi (K , L). The dual mixed volumes Vi (K , B) iswritten as Wi (K ) and V (K , . . . , K︸ ︷︷ ︸

n−i−1

, B, . . . , B︸ ︷︷ ︸i

, L) is written as Wi (K , L). The mixed vol-

ume of K1 ∩ Eu, . . . , Kn−1 ∩ Eu in (n − 1)-dimensional space will be denoted by v(K1 ∩Eu, . . . , Kn−1 ∩ Eu). If K1 = · · · = Kn−1−i = K and Kn−i = · · · = Kn−1 = L ,then v(K1 ∩ Eu, . . . , Kn−1 ∩ Eu) is written vi (K ∩ Eu, L ∩ Eu). If L = B, then vi (K ∩ Eu,

B ∩ Eu) is written wi (K ∩ Eu).

For K1, . . . , Kn ∈ ϕn , the dual mixed volume of K1, . . . , Kn , Vi (K1, . . . , Kn), is definedby Lutwak [20],

V (K1, . . . , Kn) = 1

n

Sn−1ρ(K1, u) · · · ρ(Kn, u)dS(u). (1.1.3)

From above identity , if K ∈ ϕn , i ∈ R, then

Wi (K ) = 1

n

Sn−1ρ(K , u)n−i dS(u). (1.1.4)

1.2 Intersection bodies and mixed intersection bodies

For K ∈ ϕn , there is a unique star body I K whose radial function satisfies for u ∈ Sn−1,

ρ(I K , u) = v(K ∩ Eu), (1.2.1)

It is called the intersection bodies of K . From a result of Busemann, it follows that I Kis a convex if K is convex and centrally symmetric with respect to the origin. Clearly anyintersection body is center.

The volume of intersection bodies is given by

V (I K ) = 1

n

Sn−1v(K ∩ Eu)ndS(u).

Busemann [1] proved that

V (I K )V (K )1−n ≤ ωnn−1ω

2−nn

for K ∈ Kn , with for n ≥ 3 if and only if K is an ellipsoid with center at the origin. It iscalled the Busemann intersection inequality.

The mixed intersection bodies of K1, . . . , Kn−1 ∈ ϕn , I (K1, . . . , Kn−1), whose radialfunction is defined by

ρ(I (K1, . . . , Kn−1), u) = v(K1 ∩ Eu, . . . , Kn−1 ∩ Eu), (1.2.2)

where v is (n − 1)-dimensional dual mixed volume. Moreover, If K1, . . . , Kn−1 are convex,�(K1, . . . , Kn−1), is the well known mixed projection bodies.

If K ∈ ϕn with ρ(K , u) ∈ C(Sn−1), and i ∈ R is positive, the intersection body of orderi of K is the centered star body Ii K such that [28]

ρ(Ii K , u) = 1

n − 1

Sn−1ρ(K , v)n−1−i dS(v),

for v ∈ Sn−1, where Ii K = I (K , . . . , K︸ ︷︷ ︸n−i−1

, B, . . . , B︸ ︷︷ ︸i

).

If K1 = · · · = Kn−i−1 = K , Kn−i = · · · = Kn−1 = L , then I (K1, . . . , Kn−1) is writtenas Ii (K , L). If L = B, then Ii (K , L) is written as Ii K is called the i th intersection body ofK . For I0 K simply write I K . The term is introduced by Zhang [28].

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The following property holds: If K , L and M ∈ ϕn , and λ,µ > 0, then

I1(M, λK + µL) = λI1(M, K ) + µI1(M, L). (1.2.3)

2 Some Lemmas

Lemma 2.1 [33] If K , L ∈ ϕn, i < n − 1, then

Wi (K + L)1/(n−i) ≤ Wi (K )1/(n−i) + Wi (L)1/(n−i), (2.1)

with equality if and only if K is a dilation of L. The inequality is reversed for i > n orn − 1 < i < n.

Lemma 2.2 [33] If K , L ∈ ϕn, i < n − 1, then

Wi (K , L)n−i ≤ Wi (K )n−i−1Wi (L), (2.2)

with equality if and only if K is a dilation of L. The inequality is reversed for i > n orn − 1 < i < n.

Lemma 2.3 [20] If K1, . . . , Kn ∈ ϕn, then

V (K1, . . . , Kn)r ≤r∏

j=1

V (K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn). (2.3)

with equality if and only if K1, . . . , Kn are all dilations of each other.

The special case r = n − 1 of inequality (2.3) is the following result.

Lemma 2.4 If K1, . . . , Kn ∈ ϕn, then

V (K1, . . . , Kn)n−1 ≤ V1(K1, Kn) · · · V1(Kn−1, Kn). (2.4)

with equality if and only if K1, . . . , Kn are all dilations of each other.

The following is an easy identity for mixed intersection bodies, which involves dual mixedvolumes, will facilitate a number of proofs given later.

Lemma 2.5 If K1, . . . , Kn−1, L1, . . . , Ln−1 ∈ ϕn, then

V (K1, . . . , Kn−1, I (L1, . . . , Ln−1)) = V (L1, . . . , Ln−1, I (K1, . . . , Kn−1)). (2.5)

With suitable modifications, the proof of Lemma 2.5 can be completed by the similar stepsas in the proof of “Lemma (8.9)” which was given by Lutwak [23], so we leave out the detail.

Some special cases of (2.5) will frequently be used: For K1 = · · · = Kn−i−1 = K , andKn−i = · · · = Kn−1 = B, Lemma 2.5 reduces to

Lemma 2.6 If K , L1, . . . , Ln−1 ∈ ϕn, and 0 ≤ i < n − 1 then

Wi (K , I (L1, . . . , Ln−1)) = V (L1, . . . , Ln−1, Ii K ). (2.6)

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A useful special case of (2.6) is that for K , L , M ∈ ϕn , and 0 ≤ i < n − 1 then

Wi (K , I1(L , M)) = V (L , . . . , L , M, Ii K ). (2.7)

In particular, when L = M , (2.7) changes to

Wi (K , I L) = V1(L , Ii K ). (2.8)

If K1 = · · · = Kn−i−1 = K , while Kn−i = · · · = Kn−1 = B, and L1 = · · · = Ln− j−1

= L , while Ln− j = · · · = Ln−1 = B, then Lemma 2.5 becomes

Lemma 2.7 If K , L ∈ ϕn, and 0 ≤ i, j < n − 1 then

Wi (K , I j L) = W j (L , Ii K ). (2.9)

Taking i = j = 0 to (2.9), (2.9) becomes V1(K , I L) = V1(L , I K ) which was given byLutwak [23].

Taking L1 = · · · = Ln−1 = B in Lemma 2.5, and in view of I (B, . . . , B) = I B =ωn−1 B, and get that

Lemma 2.8 If K1, . . . , Kn−1 ∈ ϕn, then

Wn−1(I (K1, . . . , Kn−1)) = ωn−1V (K1, . . . , Kn−1, B). (2.10)

For K1 = · · · = Kn−2 = K , and Kn−1 = L , (2.10) changes to

Wn−1(I1(K , L)) = ωn−1W1(K , L). (2.11)

Taking K1 = · · · = Kn−i−1 = K , and Kn−i = · · · = Kn−1 = B to (2.10), then (2.10)changes to

Wn−1(Ii K ) = ωn−1Wi+1(K ), i < n − 1. (2.12)

Lemma 2.9 If K , L , M ∈ ϕn, i < n − 2, then

Wi (K + L , M)n−i−1 ≤ Wi (K , M)n−i−1 + Wi (L , M)n−i−1, (2.13)

with equality if and only if K is a dilation of L.The inequality is reversed for i > n − 1 or n − 2 < i < n − 1.

From (1.1.3), (1.1.1) and Minkowski inequality for integral, we easy finish the proof ofLemma 2.9.

Lemma 2.10 if a1, b1, . . . , l1 ≥ 0, a2, b2, . . . , l2 > 0 and α + β + · · · + λ = 1, then

aα1 bβ

1 · · · lλ1 + aα2 bβ

2 · · · lλ2 ≤ (a1 + a2)α(b1 + b2)

β · · · (l1 + l2)λ, (2.14)

with equality if and only if a1/a2 = b1/b2 = · · · = l1/ l2.

3 The dual Minkowski inequality for mixed intersection bodies

The following dual Minkowski inequality for mixed intersection bodies will be established:For K , L ∈ ϕn , then

V (I1(K , L))n−1 ≤ V (I K )n−2V (I L)

with equality if and only if K and L are homothetic.In fact a general version of the dual Minkowski inequality for mixed intersection bodies

holds:

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Theorem 3.1 If K , L ∈ ϕn, and i < n then

Wi (I1(K , L))n−1 ≤ Wi (I K )n−2Wi (I L), (3.1)

with equality if and only if K is a dilation of L.

Proof When i = n − 1, (3.1) changes to the following

Wn−1(I1(K , L))n−1 ≤ Wn−1(I K )n−2Wn−1(I L).

From (2.11) and (2.12), it follows that

(ωn−1W1(K , L))n−1 ≤ (ωn−1W1(K ))n−2(ωn−1W1(L)).

Namely,

W1(K , L)n−1 ≤ W1(K )n−2W1(L).

This is just a special case i = 1 of Lemma 2.2.In the following, we assume i < n − 1.From (2.4), (2.7) and (2.8), we obtain that for M ∈ ϕn ,

Wi (M, I1(K , L))n−1 = V (K , . . . , K , L , Ii M)n−1 ≤ V1(K , Ii M)n−2V1(L , Ii M)

= Wi (M, I K )n−2Wi (M, I L)

with equality if and only if K , L and Ii M are dilations of each other.Now, apply Lemma 2.2 twice, and get

Wi (M, I1(K , L))n−1 ≤ Wi (M)(n−1)(n−i−1)/(n−i)Wi (I K )(n−2)/(n−i)Wi (I L)1/(n−i), (3.2)

with equality if and only if M, I K and I L are dilations of each other.In inequality (3.2), take I1(K , L) for M , and in view of Wi (M, M) = Wi (M), we can get

inequality (3.1) and the equality holds if and only if K and L are dilates.Moreover, we give a very interesting discussion about the equality conditions. Suppose

Wi (I1(K , L))n−1 = Wi (I K )n−2Wi (I L). (3.3)

From the equality conditions of inequality (3.2) and notice that intersection bodies are cen-tered, it follows that there exist λ,µ > 0, such that I1(K , L) = λI K = µI L and take themto (3.3), shows that λn−2µ = 1.

On the other hand, by (1.2.1) and (1.2.2), we have

v1(K ∩ Eu, L ∩ Eu) = λv(K ∩ Eu) and v1(K ∩ Eu, L ∩ Eu) = µv(L ∩ Eu).

Therefore

v1(K ∩ Eu, L ∩ Eu)n−1 = v(K ∩ Eu)n−2v(L ∩ Eu). (3.4)

From (3.4), and the equality conditions of the dual Minkowski inequality, hence, K ∩ Eu andL ∩ Eu are dilates, it follows that the equality holds if and only if K and L are dilates. Thisproves the equality conditions for i < n − 1.

The proof of Theorem 3.1 is complete. ��This is just a dual form of the following result which was given by Lutwak [24]:If K , L ∈ Kn , and 0 ≤ i < n, then

Wi (�1(K , L))n−1 ≥ Wi (�K )n−2Wi (�L),

with equality if and only if K and L are homothetic.

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4 The dual Aleksandrov–Fenchel inequality for mixed intersection bodies

The following dual Aleksandrov–Fenchel inequality for mixed intersection bodies whichwill be established: If K1, . . . , Kn−1 ∈ ϕn , 0 ≤ i < n, 1 ≤ j ≤ n − 1 and 1 ≤ r ≤ n − 1then

Wi (I (K1, . . . , Kn−1))r ≤

r∏

j=1

Wi (I (K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn−1)).

with equality if and only if K1, . . . , Kn−1 are all dilations of each other.In fact a general version of the dual Aleksandrov–Fenchel inequality for mixed intersection

bodies holds:

Theorem 4.1 If Ki and Di ∈ ϕn (i = 1, 2, . . . , n − 1), let Di (i = 1, 2, . . . , n − 1) aredilate copies of each other, then

Swi (I (K1, . . . , Kn−1), I (D1, . . . , Dn−1))r

≤r∏

j=1

Swi (I (K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn−1), I (D j , . . . , D j︸ ︷︷ ︸r

, Dr+1, . . . , Dn−1)), (4.1)

with equality if and only if Ki (i = 1, 2, . . . , n − 1) are dilates of each other.

Where Swi (K , L) denotes the dual quermassintegral sum function of star bodies K and L(see[35]).

Proof When i = n − 1, from (2.10) and in view of Lemma 2.3, it is easy to prove the result.Now, we assume i < n − 1, from (2.6), obtain that for M ∈ ϕn

Wi (M, I (K1, . . . , Kn−1)) = V (K1, . . . , Kn−1, Ii M). (4.2)

From Lemma 2.3, it follows that

V (K1, . . . , Kn−1, Ii M)r ≤r∏

j=1

V (K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn−1, Ii M). (4.3)

with equality if and only if K1, . . . , Kn−1, Ii M are all dilations of each other, it followsK1, . . . , Kn−1 are all dilations of each other.

Moreover, from (2.6), we have

V (K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn−1, Ii M) = Wi (M, I (K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn−1)).

(4.4)

From Lemma 2.2 and (4.4), we obtain that

V (K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn−1, Ii M)n−i

≤ Wi (M)n−i−1W (I (K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn−1)), (4.5)

with equality if and only if M is a dilation of I (K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn−1).

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Combine this with (4.2) and (4.3), it follows

Wi (M, I (K1, . . . , Kn−1))r(n−i)

≤ W (M)r(n−i−1)r∏

j=1

Wi (I (K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn−1)). (4.6)

Now taking M = I (K1, . . . , Kn−1) to (4.6), (4.6) changes to

Wi (I (K1, . . . , Kn−1))r ≤

r∏

j=1

Wi (I (K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn−1)). (*)

From equality conditions of (4.3) and (4.5), and in view of M = I (K1, . . . , Kn−1) and thefact

I (λ1 K1, . . . , λn−1 Kn−1) = λ1 · · · λn−1 I (K1, . . . , Kn−1), (4.7)

it follows if and only if K1, . . . , Kn−1 are all dilations of each other.Hence

Wi (I (D1, . . . , Dn−1))r =

r∏

j=1

Wi (I (D j , . . . , D j︸ ︷︷ ︸r

, Dr+1, . . . , Dn−1)). (**)

From (*), (**) and using Lemma 1.10 (in view of α = β = · · · = l = 1/r ) , we obtain that

Swi (I (K1, . . . , Kn−1), I (D1, . . . , Dn−1))

≤⎛

⎜⎝r∏

j=1

Swi (I (K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn−1), I (D j , . . . , D j︸ ︷︷ ︸r

, Dr+1, . . . , Dn−1))

⎟⎠

1/r

.

with equality if Ki (i = 1, 2, . . . , n − 1) are dilations of each other.The proof is complete. ��

Remark 4.1 In the special case where Di (i = 1, 2, . . . , n − 1) are single points, inequality(4.1) changes to

Wi (I (K1, . . . , Kn−1))r ≤

r∏

j=1

Wi (I (K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn−1)).

with equality if and only if K1, . . . , Kn−1 are all dilations of each other.This is just a dual form of the following result which was given by Lutwak [24]:If K1, . . . , Kn−1 ∈ Kn , 0 ≤ i < n, 1 ≤ j ≤ n − 1 and 1 ≤ r ≤ n − 1 then

Wi (�(K1, . . . , Kn−1))r ≥

r∏

j=1

Wi (�(K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn−1)).

Remark 4.2 Let Di (i = 1, 2, . . . , n−1) are single points and take for r = n−1, K1 = · · · =Kn− j−1 = K , Kn− j = · · · = Kn−1 = L and L = B in (4.1) and in view of I B = ωn−1 B,(4.1) changes to

If K ∈ ϕn , and 0 ≤ i < n and 0 < j < n − 1, then

Wi (I j K )n−1 ≤ ωj (n−i)n−1 ω

jn Wi (I K )n− j−1,

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with equality if and only if K is a ball.This is just a dual form of the following result which was given by Lutwak [24]:If K ∈ Kn , and 0 ≤ i < n and 0 < j < n − 1, then

Wi (� j K )n−1 ≤ ωj (n−i)n−1 ω

jn Wi (�K )n− j−1,

with equality if and only if K is a ball.In fact a general version of above inequality about mixed intersection bodies holds:

Theorem 4.2 If K ∈ ϕn, and 0 ≤ i < j < n − 1, while 0 < m < n, then

Wm(I j K )n−i−1 ≤ ω(n−m)( j−i)n−1 ω

j−in Wm(Ii K )n− j−1, (4.12)

with equality if and only if K is a ball.

Proof When m = n − 1, from (2.12), inequality (4.12) reduces to

ωj−in Wi (K )n− j ≥ W j (K )n−i , for 0 ≤ i < j < n. (4.13)

with equality if and only if K is a ball. This is just a special case of Lemma 2.3.In the following, we assume that m < n − 1.Suppose M ∈ ϕn , and from (2.9)

Wm(M, I j K ) = W j (K , Im M). (4.14)

By using again Lemma 2.3, we have

W j (K , Im M)n−i−1 ≤ Wn−1(Im M) j−i Wi (K , Im M)n− j−1. (4.15)

From (2.12) and inequality (4.13), it follows that

Wn−1(Im M) ≤ ωn−mω1/(n−m)n Wm(M)(n−m−1)/(n−m). (4.16)

with equality if and only if M is a ball.On the other hand, from (2.9) and in view of Lemma 2.2, and we get

Wi (K , Im M) = Wm(M, Ii K ) ≤ Wm(M)(n−m−1)/(n−m)Wm(Ii K )1/(n−m), (4.17)

with equality if and only if M and Ii K are dilates.Taking M = I j K to (4.14) and in view of (4.15), (4.16 and (4.17), we will get inequality

(4.12).Suppose there is equality in inequality (4.12):

Wm(I j K )n−i−1 = ω(n−m)( j−i)n−1 ω

j−in Wm(Ii K )n− j−1. (4.18)

From the equality conditions of inequalities (4.16) and (4.17), this implies that Ii K and I j Kmust be centered balls. Thus there exist λ,µ > 0, such that

Ii K = λB and I j K = µB. (4.19)

Taking (4.19) to (4.18), we have

ωj−in−1λ

n− j−1 = µn−i−1. (4.20)

Fix u ∈ Su . From (1.2.2) and (4.19), it follows that

wi (K ∩ Eu) = λ and w j (K ∩ Eu) = µ, (4.21)

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118 Geom Dedicata (2009) 141:109–122

where wi (K ∩ Eu) = v(K ∩ Eu, . . . , K ∩ Eu︸ ︷︷ ︸n−i−1

, B, . . . , B︸ ︷︷ ︸i

).

From (4.20) and (4.21), show that there is equality in inequality (4.12), between the (n−1)-dimensional Quermassintegral of K ∩ Eu , and thus K ∩ Eu is a ball, it follows that K mustbe a ball.

This is just a dual form of the following result which was given by Lutwak [24]:If K ∈ Kn, and 0 ≤ i < j < n − 1, while 0 < m < n, then

Wm(� j K )n−i−1 ≥ ω(n−m)( j−i)n−1 ω

j−in Wm(�i K )n− j−1,

with equality if and only if K is a ball. ��

5 The dual Brunn–Minkowski inequality for mixed intersection bodies

Theorem 5.1 If 0 ≤ i < n, while 0 ≤ j < n − 1, and K , L , M1, . . . , Mi , M ′1, . . . , M ′

j ∈ϕn, C = (M1, . . . , Mi ) and D = (M ′

1, . . . , M ′j ), then

Vi (I j (K + L , D), C)1/(n−i)(n− j−1) ≤ Vi (I j (K , D), C)1/(n−i)(n− j−1)

+Vi (I j (L , D), C)1/(n−i)(n− j−1).

Proof The cases of j = n − 2 and i = n − 1 are easy, only the cases where j < n − 2 andi < n − 1 need be treated.

Suppose Q ∈ ϕn , from the identity (2.5)

V (Q, . . . , Q︸ ︷︷ ︸n−i−1

, C, I j (K + L , D)) = V (K + L , . . . , K + L︸ ︷︷ ︸n− j−1

, D, Ii (Q, C)). (5.1)

Inequality (2.13) shows that

V (K +L , . . . , K + L︸ ︷︷ ︸n− j−1

, D, Ii (Q, C))1/(n− j−1)

≤ V (K , . . . , K︸ ︷︷ ︸n− j−1

, D, Ii (Q, C))1/(n− j−1) + V (L , . . . , L︸ ︷︷ ︸n− j−1

, D, Ii (Q, C))1/(n− j−1).

(5.2)

But from (2.5), follows

V (K , . . . , K︸ ︷︷ ︸n− j−1

, D, Ii (Q, C))1/(n− j−1) = V (Q, . . . , Q︸ ︷︷ ︸n− j−1

, C, Ii (K , D))1/(n− j−1),

and hence, inequality (2.3) gives

V (K , . . . , K︸ ︷︷ ︸n− j−1

, D, Ii (Q, C))1/(n− j−1)

≤ Vi (Q, C)(n−i−1)/(n−i)(n− j−1)Vi (I j (K , D), C)1/(n−i)(n− j−1). (5.3)

In exactly the same way, it can be seen that

V (L , . . . , L︸ ︷︷ ︸n− j−1

, D, Ii (Q, C))1/(n− j−1)

≤ Vi (Q, C)(n−i−1)/(n−i)(n− j−1)Vi (I j (L , D), C)1/(n−i)(n− j−1). (5.4)

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Combine (5.1), (5.2), (5.3), (5.4), and the result is

V (Q, . . . , Q︸ ︷︷ ︸n−i−1

, C, I j (K + L , D))1/(n− j−1)Vi (Q, C)−(n−i−1)/(n−i)(n− j−1)

≤ Vi (I j (K , D), C)1/(n−i)(n− j−1) + Vi (I j (L , D), C)1/(n−i)(n− j−1).

Take I j (K + L , D) for Q, and note that the left side of the last inequality reduces toVi (I j (K + L , D), C)1/(n−i)(n− j−1), which shows that the last inequality is the inequalityof the theorem. ��Remark 5.1 The most interesting case of the inequality of Theorem 5.1 is the special casewhere D = (B, . . . , B). In this case the inequality of Theorem 5.1 reads

Vi (I j (K + L), C)1/(n−i)(n− j−1) ≤ Vi (I j K , C)1/(n−i)(n− j−1)

+Vi (I j L , C)1/(n−i)(n− j−1). (5.5)

Taking for C = (B, . . . , B) in (5.5), (5.5) reduces to the following resultIf K , L ∈ ϕn , and 0 ≤ i < n, while 0 ≤ j < n − 2 then

Wi (I j (K + L))1/(n−i)(n− j−1) ≤ Wi (I j K )1/(n−i)(n− j−1) + Wi (I j L)1/(n−i)(n− j−1),

with equality if and only if K and L are dilates.

This is just dual Brunn–Minkowski inequality for intersection bodies which was estab-lished by Zhao and Leng [36].

On the other hand, we obtain that

Theorem 5.2 If K , L ∈ ϕn, λ, µ > 0 and i < n − 1, then

Wi (I (λK +µL))1/(n−i) ≤ λWi (I K )1/(n−i) + µWi (I L)1/(n−i),

with equality holds if and only if I K and I L are dilates.The inequality is reversed for i > n or n − 1 < i < n.

Where λK +µL is the radial Blaschke linear combination. The radial Blaschke linearcombination was defined by Lutwak [23], If K , L ∈ ϕn and λ,µ ≥ 0, then λ · K + µ · L , asthe star body whose radial function is given by:

ρ(λ · K +µ · L , ·)n−1 = λρ(K , ·)n−1 + µρ(L , ·)n−1. (5.6)

From (1.1.4), (5.6) and in view of Minkowski inequality for integral, the proof of Theorem 5.2can be completed by following the same steps as in the proof of Lemma 2.1 with suitablechanges and hence we omit the details.

Similarly, we also get the following result:

Theorem 5.2* If K , L ∈ Kn, λ, µ > 0 and i < n − 1, then

Wi (�(λK +µL))1/(n−i) ≥ λWi (�K )1/(n−i) + µWi (�L)1/(n−i),

with equality holds if and only if �K and �L are homothetic.

Where λK + µL is the Blaschke linear combination. The definition and property of theBlaschke linear combination see [23].

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6 Open problems

In a dual manner we pose the following conjecture:

Conjecture 6.1 If K1, . . . , Kn−1 ∈ ϕn, then

V (I (K1, . . . , Kn−1)) ≥ 2n−1V (K1) · · · V (Kn−1), (6.1)

with equality if and only if Ki (i = 1, 2, . . . , n − 1) are a direct sum of (centrally symmetric)planar bodies.

Remark 6.1 Taking for K1 = · · · = Kn−1 = K in (6.1), (6.1) reduces to

V (I K ) ≥ 2n−1V (K )n−1, (6.2)

with equality if and only if K is a direct sum of (centrally symmetric) planar bodies.A inverse form of inequality (6.2) is the following famous the Busemann intersection

inequality sated in Sect. 1.

V (I K ) ≤ ωnn−1ω

2−nn V (K )n−1

for K ∈ Kn , with for n ≥ 3 if and only if K is an ellipsoid with center at the origin.A dual form of inequality (6.2) is the following conjecture which was posed by Petty.

Conjecture 6.1* If K ∈ Kn, then

V (�K ) ≤ 2n V (K )n−1,

with equality if and only if K is a direct sum of (centrally symmetric) planar bodies.

Conjecture 6.2 If K1, . . . , Kn−1 ∈ ϕn and I ∗(K1, . . . , Kn−1) is the polar of mixed inter-section body of K1, . . . , Kn−1, then

V (K1) · · · V (Kn−1)V (I ∗(K1, . . . , Kn−1)) ≤ 1

n

(2nn

), (6.3)

with equality if and only if Ki (i = 1, 2, . . . , n − 1) are dilates simplexes.

Remark 6.2 Taking for K1 = · · · = Kn−1 = K in (6.3), (6.3) changes to

V (K )n−1V (I ∗(K ) ≤ 1

n

(2nn

), (6.4)

with equality if and only if K is a simplex.

A dual form of (6.4) is following result which was proved by Zhang

Theorem 6.2** If �∗K is the polar projection body of the n-dimensional convex body Kin R

n, then the inequality

V (K )n−1V (�∗K ) ≥ 1

n

(2nn

),

with equality if and only if K is a simplex.

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Geom Dedicata (2009) 141:109–122 121

Conjecture 6.3 If K1, . . . , Kn−1 ∈ ϕn, 0 ≤ i < n, 0 < j < n − 1 and 0 < r ≤ n − 1 then

Wi (I ∗(K1, . . . , Kn−1))r ≥

r∏

j=1

Wi (I ∗(K j , . . . , K j︸ ︷︷ ︸r

, Kr+1, . . . , Kn−1)). (6.5)

A dual form of (6.5) is the following result which was proved by Zhao and Leng [36].

Theorem 6.3* If K1, . . . , Kn−1 ∈ Kn, 0 ≤ i < n, 0 < j < n − 1 and 0 < r ≤ n − 1 then

Wi (�∗(K1, . . . , Kn−1))

r ≤r∏

j=1

Wi (�∗(K j , . . . , K j︸ ︷︷ ︸

r

, Kr+1, . . . , Kn−1)).

Acknowledgements The author express his grateful thanks to Professor E. Lutwak for his valuable helpduring the preparation of this paper. He provided many messages about the abstract and references. Researchis supported by Zhejiang Provincial Natural Science Foundation of China (Y605065), Foundation of the Edu-cation Department of Zhejiang Province of China (20050392), and the Academic Mainstay of Middle-age andYouth Foundation of Shandong Province of China (200203).

References

1. Busemann, H.: Volume in terms of concurrent cross-sections. Pac. J. Math. 3, 1–12 (1953)2. Campi, S.: Stability estimates for star bodies in terms of their intersection bodies. Mathematika 45, 287–

303 (1998)3. Campi, S.: Convex intersection bodies in three and four dimensions. Mathematika 46, 15–27 (1999)4. Fallert, H., Goodey, P., Weil, W.: Spherical projections and centrally symmetric sets. Adv. Math.

129, 301–322 (1997)5. Gardner, R.J.: On the Busemann–Petty problem concerning central sections of centrally symmetric

convex bodies. Bull. Am. Math. Soc. (N.S.) 30, 222–226 (1994)6. Gardner, R.J.: Intersection bodies and the Busemann–Petty problem. Trans. Am. Math. Soc. 342, 435–

445 (1994)7. Gardner, R.J.: A positive answer to the Busemann–Petty problem in three dimensions. Ann.

Math. 140, 435–447 (1994)8. Gardner, R.J.: Geometric Tomography. Cambridge University Press, Cambridge (1995)9. Gardner, R.J.: The Brunn–Minkowski inequality. Bull. Am. Math. Soc. 39, 355–405 (2002)

10. Gardner, R.J., Koldobsky, A., Schlumprecht, T.: An analytic solution to the Busemann–Petty problem.C. R. Acad. Sci. Paris Ser. I Math. 328, 29–34 (1999)

11. Gardner, R.J., Koldobsky, A., Schlumprecht, T.: An analytic solution to the Busemann–Petty problemon sections of convex bodies. Ann. Math. 149, 691–703 (1999)

12. Goodey, P., Weil, W.: Intersection bodies and ellipsoids. Mathematika 42, 295–304 (1995)13. Goodey, P., Lutwak, E., Weil, W.: Functional analytic characterizations of classes of convex bodies. Math.

Z. 222, 363–381 (1996)14. Grinberg, E., Zhang, G.: Convolutions, transforms, and convex bodies. Proc. Lond. Math. Soc. 78, 77–

115 (1999)15. Koldobsky, A.: Intersection bodies, positive definite distributions, and the Busemann–Petty prob-

lem. Am. J. Math. 120, 827–840 (1998)16. Koldobsky, A.: Second derivative test for intersection bodies. Adv. Math. 136, 15–25 (1998)17. Koldobsky, A.: Intersection bodies in R4. Adv. Math. 136, 1–14 (1998)18. Koldobsky, A.: A functional analytic approach to intersection bodies. Geom. Funct. Anal. 10, 1507–

1526 (2000)19. Koldobsky, A.: Fourier Analysis in Convex Geometry. American Mathematical Society Press, Provi-

dence, RI (2005)20. Lutwak, E.: Dual mixed volumes. Pac. J. Math. 58, 531–538 (1975)21. Lutwak, E.: Mixed projection inequalities. Trans. Am. Math. Soc. 287, 91–105 (1985)22. Lutwak, E.: Volume of mixed bodies. Trans. Am. Math. Soc. 294, 487–500 (1986)23. Lutwak, E.: Intersection bodies and dual mixed volumes. Adv. Math. 71, 232–261 (1988)24. Lutwak, E.: Inequalities for mixed projection bodies. Trans. Am. Math. Soc. 339, 901–916 (1993)

123

Page 14: On intersection and mixed intersection bodies

122 Geom Dedicata (2009) 141:109–122

25. Moszynska, M.: Quotient star bodies, intersection bodies, and star duality. J. Math. Anal. Appl. 232,45–60 (1999)

26. Zhang, G.: Intersection bodies and the four-dimensional Busemann–Petty problem. Int. Math. Res.Not. 7, 233–240 (1993)

27. Zhang, G.: Intersection bodies and the Busemann–Petty inequalities in R4. Ann. Math. 140, 331–346 (1994)

28. Zhang, G.: Centered bodies and dual mixed volumes. Trans. Am. Math. Soc. 345, 777–801 (1994)29. Zhang, G.: Sections of convex bodies. Am. J. Math. 118, 319–340 (1996)30. Zhang, G.: A positive solution to the Busemann–Petty problem in R4. Ann. Math. 149, 535–543 (1999)31. Zhang, G.: Intersection bodies and polytopes. Mathematika 46, 29–34 (1999)32. Zhao, C. J.: Extremal problems in convex bodies geometry. Dissertation for the Doctoral Degree, Shang-

hai University, Shanghai (2004)33. Zhao, C.J., Leng, G.S.: Brunn–Minkowski inequality for mixed projection bodies. J. Math. Anal.

Appl. 301, 115–123 (2005)34. Zhao, C.J., Cheung, S.W.: On p-quermassintegral differences function. Proc. Indian Acad. Sci. (Math.

Sci.) 116(2), 221–231 (2006)35. Zhao, C.J., Leng, G.S.: Inequalities for dual quermassintegrals of mixed intersection bodies. Proc. Indian

Acad. Sci. (Math. Sci.) 115(1), 79–91 (2005)36. Zhao, C.J., Leng, G.S.: On pollars of mixed projection bodies. J. Math. Anal. Appl. 316, 664–678 (2006)

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