Complex Intersection Bodies

0.27 MB·English

by  A. Koldobsky

COMPLEX INTERSECTION BODIES
A. KOLDOBSKY, G. PAOURIS, AND M. ZYMONOPOULOU
2
1
Abstract. We introduce complex intersection bodies and show
0
2 that their properties and applications are similar to those of their
real counterparts. In particular, we generalize Busemann’s the-
n
orem to the complex case by proving that complex intersection
a
J bodiesofsymmetriccomplexconvexbodiesarealsoconvex. Other

2 results include stability in the complex Busemann-Petty problem
for arbitrary measures and the corresponding hyperplane inequal-

] ity for measures of complex intersection bodies.
A
F
.
h
t
a
m
[ 1. Introduction

1
The concept of an intersection body was introduced by Lutwak [37],
v
7 aspart of his dual Brunn-Minkowski theory. Inparticular, these bodies
3
played an important role in the solution of the Busemann-Petty prob-
4
0 lem. Many results on intersection bodies have appeared in recent years
1. (see [10, 22, 34] and references there), but almost all of them apply to
0 the real case. The goal of this paper is to extend the concept of an
2
intersection body to the complex case.
1
: Let K and L be origin symmetric star bodies in Rn. Following [37],
v
i we say that K is the intersection body of L if the radius of K in every
X
direction is equal to the volume of the central hyperplane section of L
r
a perpendicular to this direction, i.e. for every ξ ∈ Sn−1,
kξk−1 = |L∩ξ⊥|, (1)
K
where kxk = min{a ≥ 0 : x ∈ aK}, ξ⊥ = {x ∈ Rn : (x,ξ) = 0}, and
K
|·| stands for volume. By a theorem of Busemann [8] the intersection
body of an origin symmetric convex body is also convex. However,
intersectionbodiesofconvexbodiesformjustasmallpartoftheclassof
intersection bodies. In particular, by results of Hensley [20] and Borell
[4], intersection bodies of symmetric convex bodies are isomorphic to
an ellipsoid, i.e. d (I(K),Bn) ≤ c where d is the Banach-Mazur
BM 2 BM
distance and c > 0 is a universal constant.
1
2 A.KOLDOBSKY,G. PAOURIS,ANDM. ZYMONOPOULOU
The right-hand side of (1) can be written using the polar formula for
volume:
1 1
kξk−1 = kθk−n+1dθ = R(k·k−n+1)(ξ),
K n−1 L n−1 L
ZSn−1∩ξ⊥
where the operator R : C(Sn−1) → C(Sn−1) is the spherical Radon
transform defined by
Rf(ξ) = f(x)dx.
ZSn−1∩ξ⊥
This means that a star body K is the intersection body of a star body
if and only if the function k·k−1 is the spherical Radon transform of a
K
continuous positive function on Sn−1.
A moregeneral class of bodies was introduced in[18]. A star bodyK
in Rn is called an intersection body if there exists a finite Borel measure
µ on the sphere Sn−1 so that k·k−1 = Rµ as functionals on C(Sn−1),
K
i.e. for every continuous function f on Sn−1,
kxk−1f(x) dx = (Rµ,f) = Rf(x) dµ(x). (2)
K
ZSn−1 ZSn−1
We introduce complex intersection bodies along the same lines. In
Section 2 we define complex intersection bodies of complex star bodies,
andinSection7we study complex intersection bodiesofconvex bodies.
While the complex version of Busemann’s theorem requires a serious
effort, the extension of the Hensley-Borell theorem to the complex case
follows from a result of Ball [2]. In Section 3 we prove that the complex
spherical Radon transform and the Fourier transform of distributions
coincide (up to a constant) on a class of (−2n+2)-homogeneous func-
tions on R2n with symmetries determined by the complex structure. A
similar result in the real case was crucial for the study of real intersec-
tion bodies. We use this result in Section 4, where we define complex
intersection bodies and prove a Fourier characterization of intersection
bodies: an origin symmetric complex star body K in R2n is a complex
intersection body if and only if the function k·k−2 represents a positive
K
definite distribution. We use this characterization in Section 5 to show
that the class of complex intersection bodies coincides with the class
of real 2-intersection bodies in R2n and, at the same time, with the
class of generalized 2-intersection bodies, provided that bodies from
the real classes possess symmetries determined by the complex struc-
ture of R2n. The latter allows to extend to the complex case a result of
Goodey and Weil [19] by showing that all symmetric complex intersec-
tion bodies can be obtained as limits in the radial metric of complex
radial sums of ellipsoids. Finally, Section 6 deals with stability in the
COMPLEX INTERSECTION BODIES 3
complex Busemann-Petty problem for arbitrary measures and related
hyperplane inequalities.
2. Complex intersection bodies of star bodies
The theory of real convex bodies goes back to ancient times and
continues to be a very active field now. The situation with complex
convex bodies is different, as no systematic studies of these bodies have
been carried out, and results appear only occasionally; see for example
[31, 35, 1, 42, 49, 50].
origin symmetric convex bodies in Cn are the unit balls of norms on
Cn. We denote by k·k the norm corresponding to the body K :
K
K = {z ∈ Cn : kzk ≤ 1}.
K
In order to define volume, we identify Cn with R2n using the standard
mapping
ξ = (ξ ,...,ξ ) = (ξ +iξ ,...,ξ +iξ ) 7→ (ξ ,ξ ,...,ξ ,ξ ).
1 n 11 12 n1 n2 11 12 n1 n2
Since norms on Cn satisfy the equality
kλzk = |λ|kzk, ∀z ∈ Cn, ∀λ ∈ C,
origin symmetric complex convex bodies correspond to those origin
symmetric convex bodies K in R2n that are invariant with respect to
any coordinate-wise two-dimensional rotation, namely for each θ ∈
[0,2π] and each ξ = (ξ ,ξ ,...,ξ ,ξ ) ∈ R2n
11 12 n1 n2
kξk = kR (ξ ,ξ ),...,R (ξ ,ξ )k , (3)
K θ 11 12 θ n1 n2 K
where R stands for the counterclockwise rotation of R2 by the angle
θ
θ with respect to the origin. We shall say that K is a complex convex
body in R2n if K is a convex body and satisfies equations (3).
Acompact set K inRn iscalled astar bodyif theoriginisaninterior
point of K, every straight line passing through the origin crosses the
boundary of K at exactly two points and the Minkowski functional of
K defined by
kxk = min{a ≥ 0 : x ∈ aK}, ∀x ∈ Rn
K
is a continuous function on Rn. The radial function of K is given by
ρ (x) = max{a > 0 : ax ∈ K}.
K
If x ∈ Sn−1, then ρ (x) is the radius of K in the direction of x. Note
K
that for any unit vector ξ, ρ (ξ) = kξk−1. The radial metric in the
K K
class of star bodies is defined by
ρ(K,L) = max |ρ (ξ)−ρ (ξ)|.
K L
ξ∈Sn−1
4 A.KOLDOBSKY,G. PAOURIS,ANDM. ZYMONOPOULOU
If the Minkowski functional of a star body K in R2n is R -invariant
θ
(i.e. satisfies equations (3)), we say that K is a complex star body in
R2n.
For ξ ∈ Cn,|ξ| = 1, denote by
n
H = {z ∈ Cn : (z,ξ) = z ξ = 0}
ξ k k
k=1
X
the complex hyperplane through the origin, perpendicular to ξ. Under
the standard mapping from Cn to R2n the hyperplane H turns into a
ξ
(2n−2)-dimensional subspace of R2n orthogonal to the vectors
ξ = (ξ ,ξ ,...,ξ ,ξ ) and ξ⊥ = (−ξ ,ξ ,...,−ξ ,ξ ).
11 12 n1 n2 12 11 n2 n1
The orthogonal two-dimensional subspace H⊥ has orthonormal basis
ξ
ξ,ξ⊥ . A star (convex) body K in R2n is a complex star (convex)
body if and only if, for every ξ ∈ S2n−1, the section K ∩ H⊥ is a
(cid:8) (cid:9) ξ
two-dimensional Euclidean circle with radius ρ (ξ) = kξk−1.
K K
We introduce complex intersection bodies of complex star bodies us-
ing a definition under which these bodies play the same role in complex
convexity, as their real counterparts in the real case. We use the nota-
tion |K| for the volume of K; the dimension where we consider volume
is clear in every particular case.
Definition 1. Let K,L be origin symmetric complex star bodies in
R2n. We say that K is the complex intersection body of L and write
K = I (L) if for every ξ ∈ R2n
c
|K ∩H⊥| = |L∩H |. (4)
ξ ξ
Since K ∩ H⊥ is the two-dimensional Euclidean circle with radius
ξ
kξk−1, (4) can be written as
K
πkξk−2 = |L∩H |. (5)
Ic(L) ξ
AllthebodiesK thatappearascomplex intersection bodiesofdifferent
complexstarbodiesformthe class of complex intersection bodies of star
bodies. In Section 4, we will introduce a more general class of complex
intersection bodies.
3. The Radon and Fourier transforms of R -invariant
θ
functions
Denote by C (S2n−1) the space of R -invariant continuous functions,
c θ
i.e. continuous real-valued functions f on the unit sphere S2n−1 in R2n
satisfying f(ξ) = f(R (ξ)) for all ξ ∈ S2n−1 and all θ ∈ [0,2π]. The
θ
COMPLEX INTERSECTION BODIES 5
complex spherical Radon transform is an operator R : C (S2n−1) →
c c
C (S2n−1) defined by
c
R f(ξ) = f(x)dx.
c
ZS2n−1∩Hξ
Writing volume in polar coordinates, we get that for every complex
star body L in R2n and every ξ ∈ S2n−1,
1 1
|L∩H | = kxk−2n+2dx = R k·k−2n+2 (ξ),
ξ 2n−2 L 2n−2 c L
ZS2n−1∩Hξ (6)
(cid:0) (cid:1)
so the condition (5) reads as
1
kξk−2 = R k·k−2n+2 (ξ). (7)
Ic(L) 2π(n−1) c L
(cid:0) (cid:1)
This means that a complex star body K is a complex intersection body
of a star body if and only if the function k·k−2 is the complex spher-
K
ical Radon transform of a continuous positive R -invariant function
θ
on S2n−1. We use this observation in Section 4, where we introduce a
more general class of complex intersection bodies (not depending on
the underlying star body), like it was done in the real case in [18]. But
before that we need several facts connecting the Radon transform to
the Fourier transform in the complex setting.
We use the techniques of the Fourier transform of distributions; see
[14] for details. As usual, we denote by S(Rn) the Schwartz space of
rapidly decreasing infinitely differentiable functions (test functions) in
Rn, and S′(Rn) is the space of distributions over S(Rn).
Suppose that f is a locally integrable complex-valued function on Rn
with power growth at infinity, i.e. there exists a number β > 0 so that
f(x)
lim = 0,
|x|2→∞ |x|β2
where |·| stands for the Euclidean norm on Rn. Then f represents a
2
distribution acting by integration: for every φ ∈ S,
hf,φi = f(x)φ(x) dx.
Rn
Z
ˆ ˆ
The Fouriertransformofadistributionf isdefined byhf,φi = hf,φi
for every test function φ. If φ is an even test function, then (φˆ)∧ =
(2π)nφ, so the Fourier transform is self-invertible (up to a constant)
for even distributions.
A distribution f is called even homogeneous of degree p ∈ R if
hf(x),φ(x/α)i = |α|n+phf,φi
6 A.KOLDOBSKY,G. PAOURIS,ANDM. ZYMONOPOULOU
foreverytestfunctionφandeveryα ∈ R, α 6= 0.TheFouriertransform
of an even homogeneous distribution of degree p is an even homoge-
neous distribution of degree −n−p.
We say that a distribution is positive definite if its Fourier transform
ˆ
is a positive distribution in the sense that hf,φi ≥ 0 for every non-
negative test function φ. Schwartz’s generalization of Bochner’s theo-
rem (see, for example, [15, p.152]) states that a distribution is positive
definite if and only if it is the Fourier transform of a tempered measure
on Rn. Recall that a (non-negative, not necessarily finite) measure µ
is called tempered if
(1+|x| )−β dµ(x) < ∞
2
Rn
Z
for some β > 0.
Our definition of a star body K assumes that the origin is an interior
point of K. If 0 < p < n, then k · k−p is a locally integrable function
K
on Rn and represents an even homogeneous of degree −p distribution.
If k·k−p represents a positive definite distribution for some p ∈ (0,n),
K
then its Fourier transform is a tempered measure which is at the same
time a homogeneous distribution of degree −n + p. One can express
such a measure in polar coordinates, as follows.
Proposition 3.1. ([22, Corollary 2.26]) Let K be an origin symmetric
convex body in Rn and p ∈ (0,n). The function k · k−p represents a
K
positive definite distribution on Rn if and only if there exists a finite
Borel measure µ on Sn−1 so that for every even test function φ,
∞
kxk−pφ(x) dx = tp−1φˆ(tξ)dt dµ(ξ).
K
ZRn ZSn−1(cid:18)Z0 (cid:19)
For any even continuous function f on the sphere Sn−1 and any non-
zero number p ∈ R, we denote by f ·rp the extension of f to an even
homogeneous function of degree p on Rn defined as follows. If x ∈ Rn,
then x = rθ, where r = |x| and θ = x/|x| . We put
2 2
f ·rp(x) = f (θ)rp.
It was proved in [22, Lemma 3.16] that, for any p ∈ (−n,0) and in-
finitely smooth function f on Sn−1, the Fourier transform of f ·r−p is
equal to another infinitely smooth function h on Sn−1 extended to an
even homogeneous of degree −n+p function h·r−n+p on the whole of
Rn. The following Parseval formula on the sphere was proved in [22,
Corollary 3.22].
COMPLEX INTERSECTION BODIES 7
Proposition 3.2. Let f,g be even infinitely smooth functions on Sn−1,
and p ∈ (0,n). Then
(f ·r−p)∧(θ)(g ·r−n+p)∧(θ) = (2π)n f(θ)g(θ) dθ.
ZSn−1 ZSn−1 (8)
We need a simple observation that will, however, provide the basis
for applications of the Fourier transform to complex bodies.
Lemma 3.3. Suppose that f ∈ C (S2n−1) is an even infinitely smooth
c
function. Then for every 0 < p < 2n and ξ ∈ S2n−1 the Fourier
transform of the distribution f ·r−p is a constant function on S2n−1 ∩
H⊥.
ξ
Proof. By [22, Lemma 3.16], the Fourier transform of f·r−p is a con-
tinuousfunctionoutsideoftheorigininR2n.Thefunctionf isinvariant
with respect to all R , so by the connection between the Fourier trans-
θ
form of distributions and linear transformations, the Fourier transform
of f ·r−p is also invariant with respect to all R . Recall that the two-
θ
dimensional space H⊥ is spanned by vectors ξ and ξ⊥ (see the Intro-
ξ
duction). Every vector in S2n−1 ∩ H⊥ is the image of ξ under one of
ξ
the coordinate-wise rotations R , so the Fourier transform of f ·r−p is
θ
a constant function on S2n−1 ∩H⊥.
ξ
✷
The following connection between the Fourier and Radon transforms
is well-known; see for example [22, Lemma 3.24].
Proposition 3.4. Let 1 ≤ k < n, and let φ ∈ S(Rn) be an even test
function. Then for any (n−k)-dimensional subspace H of Rn
1
ˆ
φ(x)dx = φ(x)dx.
(2π)k
ZH ZH⊥
We also use the spherical version of Proposition 3.4; see [22, Lemma
3.25].
Proposition 3.5. Let φ be an even infinitely smooth function on Sn−1,
let 0 < k < n, and let H be an arbitrary (n−k)-dimensional subspace
of Rn. Then
1
φ(θ)dθ = φ·r−n+k ∧(θ)dθ.
(2π)k
ZSn−1∩H ZSn−1∩H⊥
(cid:0) (cid:1)
Let us translate the latter fact to the complex situation.
8 A.KOLDOBSKY,G. PAOURIS,ANDM. ZYMONOPOULOU
Lemma 3.6. Let φ ∈ C (S2n−1) be an even infinitely smooth function.
c
Then for every ξ ∈ S2n−1
1
R φ(ξ) = φ·r−2n+2 ∧(ξ).
c
2π
Proof. By Proposition 3.5, (cid:0) (cid:1)
1
R φ(ξ) = φ(θ)dθ = φ·r−2n+2 ∧(θ)dθ,
c
ZS2n−1∩Hξ (2π)2 ZS2n−1∩Hξ⊥
(cid:0) (cid:1)
and, by Lemma 3.3, the function under the integral in the right-hand
sideisconstantonS2n−1∩H⊥.Thevalueofthisconstantisthefunction
ξ
value at ξ ∈ S2n−1 ∩ H⊥. Also, recall that S2n−1 ∩ H⊥ is the two-
ξ ξ
dimensional Euclidean unit circle, so
φ·r−2n+2 ∧(θ)dθ = 2π φ·r−2n+2 ∧(ξ).
ZS2n−1∩Hξ⊥
(cid:0) (cid:1) (cid:0) (cid:1) ✷
Lemma 3.7. The complex spherical Radon transform is self-dual, i.e.
for any even functions f,g ∈ C (S2n−1),
c
R f(ξ)g(ξ)dξ = f(θ)R g(θ)dθ.
c c
ZS2n−1 ZS2n−1
Proof. By approximation, it is enough to consider the case where
f,g are infinitely smooth. For some infinitely smooth even function
h ∈ C (S2n−1), we have g · r−2n+2 = (h·r−2)∧, then (g ·r−2n+2)∧ =
c
(2π)2nh·r−2. By Lemma 3.6 and the spherical Parseval formula (8),
1
R f(ξ)g(ξ)dξ = f ·r−2n+2 ∧(ξ)(g·r−2n+2)(ξ)dξ
c
2π
ZS2n−1 ZS2n−1
(cid:0) (cid:1)
(2π)2n
= f ·r−2n+2 ∧(ξ)(h·r−2)∧(ξ)dξ
2π
ZS2n−1
1 (cid:0) (cid:1)
= f(θ) g ·r−2n+2 ∧(θ)dθ = f(θ)R g(θ)dθ.
c
2π
ZS2n−1 ZS2n−1 ✷
(cid:0) (cid:1)
We now prove Lemma 3.6 without smoothness assumption. This re-
sult isa complex version of [22, Lemma 3.7]. We say that a distribution
f on R2n is R -invariant if hf,φ(R (·))i = hf,φi for every test function
θ θ
φ ∈ S(R2n) and every θ ∈ [0,2π]. If f and g are R -invariant distribu-
θ
tions, and hf,φi = hg,φi for any test function φ that is invariant with
respect to all R , then f = g. This follows from the observation that
θ
the value of an R -invariant distribution on a test function φ does not
θ
change if φ is replaced by the function 1 2πφ(R (·))dθ.
2π 0 θ
R
COMPLEX INTERSECTION BODIES 9
Lemma 3.8. Let f ∈ C (S2n−1) be an even function. Then the Fourier
c
transform of f ·r−2n+2 is a continuous function on the sphere extended
to a homogeneous function of degree -2 on the whole R2n. Moreover,
on the sphere this function is equal (up to a constant) to the complex
spherical Radon transform of f: for any ξ ∈ S2n−1,
1
R f(ξ) = f ·r−2n+2 ∧(ξ).
c
2π
Proof. Let φ ∈ S(R2n) be any e(cid:0)ven R -inv(cid:1)ariant test function. Then
θ
ˆ
φ is also an even R -invariant test function. By Lemma 3.3, for any
θ
ξ ∈ S2n−1,
∞ ∞
ˆ ˆ ˆ
φ(x)dx = rφ(rθ)dr dθ = 2π rφ(rξ)dr.
ZHξ⊥ ZS2n−1∩Hξ⊥ (cid:18)Z0 (cid:19) Z0 (9)
By Proposition 3.4 and Lemma 3.7,
h f ·r−2n+2 ∧,φi = |x|−2n+2f(x/|x| )φˆ(x)dx
2 2
R2n
Z
(cid:0) (cid:1)
∞ 1
ˆ ˆ
= f(ξ) rφ(rξ)dr dξ = f(ξ) φ(x)dx dξ
ZS2n−1 (cid:18)Z0 (cid:19) 2π ZS2n−1 ZHξ⊥ !
= 2π f(ξ) φ(x)dx dξ
ZS2n−1 ZHξ !
∞
= 2π f(ξ) r2n−3φ(rθ)dr dθ dξ
ZS2n−1 ZS2n−1∩Hξ (cid:18)Z0 (cid:19) !
∞
= 2π f(ξ) R r2n−3φ(r·)dr (ξ)dξ
c
ZS2n−1 (cid:18)Z0 (cid:19)
∞
= 2π r2n−3φ(rθ)dr R f(θ)dθ
c
ZS2n−1 (cid:18)Z0 (cid:19)
= 2π |x|−2R f(x/|x| )φ(x)dx.
2 c 2
R2n
Z
We get that for every even R -invariant test function φ,
θ
h f ·r−2n+2 ∧,φi = 2πh|x|−2R f(x/|x| ),φi,
2 c 2
soevenR -inv(cid:0)ariantdistr(cid:1)ibutions(f ·r−2n+2)∧ and2π|x|−2R f(x/|x| )
θ 2 c 2
are equal.
✷
10 A.KOLDOBSKY,G. PAOURIS,ANDM. ZYMONOPOULOU
Lemma 3.8 implies the following Fourier transform formula for the
volume of sections of star bodies. Note that the real version of this
formula was proved in [23], and that the complex formula below was
proved in [31] for infinitely smooth bodies by a different method; here
we remove the smoothness condition.
Theorem 3.9. Let K be an origin symmetric complex star body in
R2n,n ≥ 2. For every ξ ∈ S2n−1, we have
1
|K ∩H | = kxk−2n+2 ∧(ξ).
ξ 4π(n−1) K
(cid:0) (cid:1)
Proof. By(6)andLemma3.8appliedtothefunctionf(θ) = kθk−2n+2,
K
1 1
|K ∩H | = R k·k−2n+2 (ξ) = kxk−2n+2 ∧(ξ).
ξ 2n−2 c K 4π(n−1) K
(cid:0) (cid:1) (cid:0) (cid:1) ✷
We use Theorem 3.9 to prove the complex version of the Minkowski-
Funk theorem saying that an origin symmetric star body is uniquely
determined by volume of its central hyperplane sections; see [22, Corol-
lary 3.9].
Corollary 3.10. If K,L are origin symmetric complex star bodies in
R2n, and their intersection bodies I (K) and I (L) coincide, then K =
c c
L.
Proof. The equality of intersection bodies means that, for every
ξ ∈ S2n−1,|K∩H | = |L∩H |.ByTheorem3.9, homogeneous ofdegree
ξ ξ
-2 continuous on R2n \ {0} functions k·k−2n+2 ∧ and k·k−2n+2 ∧
K L
coincide on the sphere S2n−1, so they are also equal as distributions on
(cid:0) (cid:1) (cid:0) (cid:1)
the whole of R2n. The result follows from the uniqueness theorem for
the Fourier transform of distributions.
✷
4. Complex intersection bodies
We are going to define the class of complex intersection bodies by
extending the equality (7) to measures, as it was done in the real case
in [18]. We say that a finite Borel measure µ on S2n−1 is R -invariant
θ
if for any continuous function f on S2n−1 and any θ ∈ [0,2π],
f(x)dµ(x) = f(R x)dµ(x).
θ
ZS2n−1 ZS2n−1