ON THE LOCAL EQUATORIAL CHARACTERIZATION OF ZONOIDS AND INTERSECTION BODIES

FEDOR NAZAROV, DMITRY RYABOGIN AND ARTEM ZVAVITCH

Abstract. In this paper we show that there is no local equatorial characterization of zonoids in odd dimensions. This gives a negative answer to the conjecture posed by W. Weil in 1977 and shows that the local equatorial characterization of zonoids may be given only in even dimensions. In addition we prove a similar result for intersection bodies and show that there is no local characterization of these bodies.

1. Introduction.

A zonoid in ℝ^{n} is an origin symmetric convex body that can be approximated (in the Hausdorff metric) by finite Minkowski sums of line segments. It turns out that zonoids appear in many different contexts in convex geometry, physics, optimal control theory, and functional analysis (we refer the reader to [B], [BL], [BLM], [Ga2], [GW2], [P], [Sc1], [Sc2], [ScW]). One of the equivalent definitions of zonoids, useful in convex geometry, leads to a notion of a projection body. An origin symmetric convex body L in ℝ^{n} is called a projection body if there exists another origin symmetric convex body K such that the support function of L in every direction is equal to the volume of the hyperplane projection of K orthogonal to this direction: for every ξ ∈ S^{n−1},

h_{L}(ξ) = Vol_{n−1}(K|ξ^{⊥}),

ξ^{⊥} = {y ∈ ℝ^{n} : ξ · y = 0}. The support function h_{L}(ξ) = max_{x∈L} ξ · x is equal to the dual norm ∥ξ∥_{L^{∗}} where L^{∗} stands for the polar body of L. From the above definition and Cauchy formula (see [K], page 25), we immediately derive the following analytic definition, which will be useful for us in this paper: An origin symmetric convex body L ⊂ ℝ^{n} is a zonoid if and only if

h_{L}(ξ) = Cosμ(ξ) := ∫_{S^{n−1}} |ξ · θ|dμ(θ)

with some even positive measure μ on S^{n−1}. Finally, a functional analytic definition shows that an origin symmetric convex body L ⊂ ℝ^{n} is a zonoid if and only if it is a polar body to the unit ball of a subspace of L_{1}.

Date: December 3, 2007.

1991 Mathematics Subject Classification. Primary 52A15, 52A21.

Key words and phrases. Convex bodies, Radon Transform, Cosine Transform, Fourier Transform.

Authors were partially supported by Kansas State NSF-CBMS grant DMS0532656. The first author was partially supported by the NSF grant DMS0501067. The second author was partially supported by the NSF grant DMS0400789, the third author was partially supported by the NSF grant DMS0504049 .

It is well known that every origin symmetric convex body in ℝ^{2} is a projection body, but this is no longer true in ℝ^{n} for n ≥ 3 (see [Sc2], [K]). It is an interesting question how to determine if a given convex body is a zonoid or not. It is very reasonable to assume that one can provide a strictly local characterization of zonoids. This question was posed repeatedly (see [Sc2] for the history of the problem), however W. Weil showed [W] that a local characterization of zonoids does not exist. In particular, he showed that there exists an origin-symmetric convex C^{∞} body K ⊂ ℝ^{n}, n ≥ 3, that is not a zonoid but has the following property: for every u ∈ S^{n−1} there exists a zonoid Z_{u} centered at the origin and a neighborhood U_{u} ⊂ S^{n−1} of u such that the boundaries of K and Z_{u} coincide at all points where the exterior unit normal vectors belong to U_{u}. Thus, no characterization of zonoids that involves only arbitrarily small neighborhoods of boundary points is possible.

In 1977, W. Weil (see [W]) proposed the following conjecture about local equatorial characterization of zonoids. Let L ⊂ ℝ^{n} be an origin-symmetric convex body and assume that for any equator σ ⊂ S^{n−1}, there exists a zonoid Z_{σ} and a neighborhood E_{σ} of σ such that the boundaries of L and Z_{σ} coincide at all points where the exterior unit vector belongs to E_{σ}; then L is a zonoid. Affirmative answers for even dimensions were given independently by G. Panina [Pan] in 1988 and Goodey and Weil [GW] in 1993, but the question was left open in odd dimensions. That was a consequence of the fact that the inversion formulas for the cosine transform are not local in odd dimensions.

In this paper we show that the answer to the conjecture in odd dimensions is negative. We prove that in both cases (for odd and even dimensions) the answer can be obtained as a consequence of the characterization of zonoids in terms of sections of the polar body, given in [KRZ]. In even dimensions the answer follows directly from the geometric inversion formula for the Cosine transform [KRZ]. The odd dimensional case, on the other hand, requires much more tricky and detailed analysis of the behavior of the inverse Cosine transform.

Our main tool is the Fourier analytic inversion formula from [GKS2] (see equation (3), (4) below or [K], page 60). It allows to obtain the results for zonoids together with the results about the intersection bodies. The notion of an intersection body of star body was introduced by E. Lutwak [Lu]. K is called the intersection body of L if the radius of K in every direction is equal to the (n − 1)-dimensional volume of the central hyperplane section of L perpendicular to this direction: ∀ξ ∈ S^{n−1},

ρ_{K}(ξ) = Vol_{n−1}(L ∩ ξ^{⊥}),

where ρ_{K}(ξ) = max{a : aξ ∈ K} is the radial function of the body K. Passing to polar coordinates in ξ^{⊥}, we derive the following analytic definition of an intersection body of star body: K is called the intersection body of L if

ρ_{K}(ξ) = \frac{1}{n − 1}ℜρ_{L}^{n−1}(ξ) := \frac{1}{n − 1} ∫_{S^{n−1}∩ξ^{⊥}} ρ_{L}^{n−1}(θ)dθ.

Here ℜ stands for the spherical Radon transform.

A more general class of intersection bodies was defined by R. Gardner [Ga1] and G. Zhang [Zh] as the closure of intersection bodies of star bodies in the radial metric d(K, L) = sup_{ξ∈S^{n−1}} |ρ_{K}(ξ) − ρ_{L}(ξ)|. In this paper we will consider only C^{∞} smooth intersection bodies: a body K is an intersection body if there exists an even nonnegative function f on S^{n−1}, such that the radial function of K is a spherical Radon transform ℜf of f. Since we can always define L : ρ_{L}^{n−1}(θ) = (n − 1)f(θ), we will not distinguish between intersection bodies of star bodies and intersection bodies.

We prove that the local equatorial characterization of intersection bodies is not possible in odd dimensions. Namely, we show that one can construct an origin-symmetric convex body L ⊂ ℝ^{n}, n ≥ 5 is odd, such that for any equator σ ⊂ S^{n−1}, there exists an intersection body I_{σ} and a neighborhood E_{σ} of σ such that the boundaries of L and I_{σ} coincide at all points of E_{σ} (i.e. ρ_{L}(ξ) = ρ_{I_{σ}}(ξ) for all ξ ∈ E_{σ}); but nevertheless, L is not an intersection body. On the other hand, we show that the local equatorial characterization of intersection bodies is possible in even dimensions.

We also extend the result of W. Weil [W] to the class of intersection bodies by proving that there is no local characterization of those bodies in odd and even dimensions. We prove that there exists an origin-symmetric convex C^{∞} body K ⊂ ℝ^{n}, n ≥ 5, that is not an intersection body, but has the following property: for each u ∈ S^{n−1} there exists an intersection body I_{u} centered at the origin and a neighborhood U_{u} ⊂ S^{n−1} of u such that the boundaries of K and I_{u} coincide on U_{u}. In odd dimensions this is a consequence of the lack of a local equatorial characterization of intersection bodies mentioned above but we give an independent proof that does not distinguish between even and odd dimensions.

Our proofs for zonoids and intersection bodies are very similar, they are based on almost identical Fourier analytic inversion formulas for the Cosine and Radon transforms. This is one more indication of the remarkable duality between sections and projections (see [KRZ1]). Acknowledgement: The authors are very grateful to Paul Goodey, Alexander Koldobsky, Jeffrey Schlaerth and Wolfgang Weil for many useful discussions.

2. Auxiliary results

Our main tool is the Fourier transform of distributions (see [GS], [GV] and [K] for exact definitions and properties) and the connections between the Cosine and the spherical Radon transforms and the Fourier transform.

We start with the connection of the spherical Radon transform and the Fourier transform. A. Koldobsky (see, for example, [K], Lemma 3.7) proved that

(1)  ℜg(ξ) = \frac{1}{π} ĝ(ξ), ∀ξ ∈ S^{n−1},

provided that g is an even homogeneous function of degree −n+1 on ℝ^{n} \{0}, n > 1, satisfying g|_{S^{n−1}} ∈ L_{1}(S^{n−1}).

An immediate consequence of this formula is the following Fourier analytic characterization of intersection bodies (see [K], Theorem 4.1): An origin-symmetric star body K is an intersection body if and only if ρ_{K}, extended to ℝ^{n} as a homogenous function of degree −1, represents a positive definite distribution on ℝ^{n}. When K is infinitely smooth, this is equivalent to ρ̂_{K} ≥ 0.

A very similar connection of the Cosine transform and the Fourier transform was established in [KRZ] (see also [K], page 155):

(2)  Cosg(ξ) = −\frac{2}{π} ĝ(ξ), ∀ξ ∈ S^{n−1},

provided that g is an even homogeneous function of degree −n−1 on ℝ^{n} \{0}, n > 1, satisfying g|_{S^{n−1}} ∈ L_{1}(S^{n−1}).

As above, one can obtain a very similar Fourier analytic characterization of zonoids (see [K], Theorem 8.6): An origin-symmetric star body K is a zonoid if and only if h_{K}, extended to ℝ^{n} as a homogenous function of degree 1, represents a negative definite distribution on ℝ^{n}. When K is infinitely smooth, this is equivalent to ĥ_{K} ≤ 0.

Our next tool is a formula connecting the Fourier transform of powers of the radial function with the derivatives of the parallel section function. Let D be an infinitely smooth origin symmetric star body in ℝ^{n}, ξ ∈ S^{n−1}, and let ξ^{⊥} = {x ∈ ℝ^{n} : x·ξ = 0}. We denote by

A_{D,ξ}(t) = Vol_{n−1}(D ∩ {ξ^{⊥} + tξ}), t ∈ ℝ,

the parallel section function of D in the direction of ξ. The following formula was proved in [GKS2] (see [K], page 60): For any ξ ∈ S^{n−1} and k ∈ ℕ, k ≠ n − 1,

(3)  { \atop ρ_{D}^{n−k−1}}(ξ) = (−1)^{k/2}π(n − k − 1)A_{D,ξ}^{(k)}(0),

when k is even, and

(4) { \atop ρ_{D}^{n−k−1}}(ξ)=(−1)^{\frac{k+1}{2}} 2(n − k − 1)k!∫_{0}^{∞} \frac{A_{D,ξ}(z) − A_{D,ξ}(0) − ... − A_{D,ξ}^{(k−1)}(0)\frac{z^{k−1}}{(k−1)!}}{z^{k+1}} dz,

when k is odd.

As a consequence of equations (1), (3), and (4) with k = n − 2, we obtain the Fourier analytic characterization of intersection bodies (see [K], page 74 for more details).

Let L be an origin symmetric star body in ℝ^{n} such that ρ_{L} is infinitely differentiable on S^{n−1}. The body L is an intersection body if and only if ∀ξ ∈ S^{n−1},

(5)  (−1)^{(n−2)/2}A_{L,ξ}^{(n−2)}(0) ≥ 0,

when n is even, and

(6)  (−1)^{(n−1)/2} ∫_{0}^{∞} \frac{A_{L,ξ}(z) − A_{L,ξ}(0) − ... − A_{L,ξ}^{(n−3)}(0)\frac{z^{n−3}}{(n−3)!}}{z^{n−1}} dz ≥ 0,

when n is odd.

Similarly, using the duality relation h_{D} = ρ_{D^{∗}}^{−1} and equations (2), (3), and (4) with k = n, one can obtain the following characterization of zonoids (see [KRZ], or [K], page 156):

Let L be an origin symmetric convex body in ℝ^{n} such that h_{L} is infinitely differentiable on S^{n−1}. The body L is a zonoid (projection body) if and only if ∀ξ ∈ S^{n−1},

(7)  (−1)^{n/2}A_{L^{∗},ξ}^{(n)}(0) ≥ 0,

when n is even, and

(8)  (−1)^{(n+1)/2} ∫_{0}^{∞} \frac{A_{L^{∗},ξ}(z) − A_{L^{∗},ξ}(0) − ... − A_{L^{∗},ξ}^{(n−1)}(0)\frac{z^{n−1}}{(n−1)!}}{z^{n+1}} dz ≥ 0,

when n is odd.

3. There is no local equatorial characterization of intersection bodies in odd dimensions.

To construct a counterexample, it is natural to use (6). This formula shows that one has to use the information about the section function A_{L,ξ}(z) of the body along the whole range of z.

For 0 < ε < 1 and ξ ∈ S^{n−1}, we denote by U_{ε}(ξ) the union of caps centered at ξ and −ξ:

U_{ε}(ξ) := \{θ ∈ S^{n−1} : |θ · ξ| ≥ \sqrt{1 − ε^{2}}\}.

We denote by E_{ε}(ξ), 0 < ε < 1, the neighborhood of the equator S^{n−1} ∩ ξ^{⊥}:

E_{ε}(ξ) := {θ ∈ S^{n−1} : |θ · ξ| < ε}.

The following result is crucial for the construction of the counterexample. Its proof is based on the fact that the inversion formula (6) is not local.

Lemma 3.1. Let n ≥ 3 be odd. Then there exists an ε > 0 and an absolute constant c > 0 such that for any x, ξ ∈ S^{n−1}, there exists an even function f_{x,ξ} satisfying f_{x,ξ} = 0 on E_{ε}(x), and ℜ^{−1}f_{x,ξ} ≥ c on U_{ε}(ξ).

Proof. First, we fix x, ξ ∈ S^{n−1} and find ε = ε(x, ξ) and c = c(x, ξ) satisfying the requirement of the lemma. Then we use the compactness argument to produce absolute ε and c.

For fixed x, ξ ∈ S^{n−1} and some small ε > 0 we take two auxiliary infinitely smooth symmetric star bodies M, Q, such that ρ_{M} = ρ_{Q} on the closure of E_{ε}(ξ) ∪ E_{ε}(x), and ρ_{M} > ρ_{Q} otherwise. We put f_{x,ξ} = (−1)^{(n−1)/2}(ρ_{M} − ρ_{Q}). Then f_{x,ξ} = 0 on E_{ε}(x), and ρ_{M} = ρ_{Q} on E_{ε}(ξ) implies A_{M,ξ}^{(k)}(0) = A_{Q,ξ}^{(k)}(0), k = 0, 1, ..., n − 3. Thus, (1) and (4) with k = n − 2 imply

ℜ^{−1}f_{x,ξ}(ξ) = (−1)^{(n−1)/2}(ℜ^{−1}ρ_{M}(ξ) − ℜ^{−1}ρ_{Q}(ξ)) =

(−1)^{n−1}(2π)^{1−n}(n − 2)! ∫_{0}^{∞} \frac{A_{M,ξ}(z) − A_{Q,ξ}(z)}{z^{n−1}} dz > 0,

since Q ⊆ M. We proved that for fixed x, ξ ∈ S^{n−1} there exists ε′ = ε′(x, ξ) > 0 and c′ = c′(x, ξ) such that there exists an even function f_{x,ξ} satisfying f_{x,ξ} = 0 on E_{ε}(x), and ℜ^{−1}f_{x,ξ}(ξ) ≥ c′.

The function ℜ^{−1}f_{x,ξ} is continuous on S^{n−1} since M, Q are infinitely smooth (see Lemma 2.4, [K]). Hence, ℜ^{−1}f_{x,ξ} ≥ c > 0 on U_{ε′′}(ξ), for some ε′′ > 0 and c = c(x, ξ).

Put ε̃ = ε̃(x, ξ) = min(ε′, ε′′). We prove that for any x and ξ, there is ε̃ = ε̃(x, ξ) > 0 and a function f_{x,ξ} such that f_{x,ξ} = 0 on Eε̃(x), but ℜ^{−1}f_{x,ξ} ≥ c on Uε̃(ξ), c = c(x, ξ).

Now we use the compactness argument to show that we can choose ε and c independent of x and ξ. We choose a finite set of pairs {x_{i}, ξ_{i}}_{i=1}^{m} such that {U_{ε̃_{i}/2}(x_{i}) × U_{ε̃_{i}/2}(ξ_{i})}_{i=1}^{m} cover S^{n−1} × S^{n−1}. We take

ε = \frac{1}{2} min_{1≤i≤m} ε̃_{i} and c = min_{1≤i≤m} c(x_{i}, ξ_{i}).

Then, for any (x, ξ), there is a pair (x_{i}, ξ_{i}) such that (x, ξ) ∈ U_{ε̃_{i}/2}(x_{i}) × U_{ε̃_{i}/2}(ξ_{i}) and thereby

E_{ε}(x) × U_{ε}(ξ) ⊂ Eε̃_{i}(x_{i}) × Uε̃_{i}(ξ_{i}).

Finally, we may define f_{x,ξ} = f_{x_{i},ξ_{i}}.

□

Remark 3.2. Note that, dilating M and Q (and thus functions f_{x,ξ}), we may assume that c is as large as we want. By the technical reasons that will become clear later, we take c = 2ℜ^{−1}1. Moreover, we can assume that the set of functions {f_{x,ξ}}_{x,ξ∈S^{n−1}} in the lemma is finite.

Let C_{+}^{∞} be the class of origin-symmetric convex bodies with C^{∞} boundary and everywhere positive Gaussian curvature (see [Ga2], page 25). The following auxiliary result seems to be well-known. It is interesting to note that it is not true without the C_{+}^{∞} assumption though (see [Sc2], pages 117, 118, and [Ki], [KP], [Bo]).

Lemma 3.3. Let M ∈ C_{+}^{∞} and let K(t) = tB_{2}^{n} + (1 − t)M be the Minkowski sum of tB_{2}^{n} and (1−t)M, t ∈ [0, 1]. Then the map t → ℜ^{−1}ρ_{K(t)}(ξ), ξ ∈ S^{n−1}, is continuous.

Proof. We note first that for any fixed t ∈ [0, 1], the boundary ∂K(t) of K(t) is C^{∞}. Indeed, ∂K(t) can be parameterized as

u ∈ S^{n−1} → ∇h_{(1−t)M}(u) + tu = (1 − t)∇h_{M}(u) + tu,

where u ∈ S^{n−1} → (1 − t)∇h_{M}(u) is a parametrization of (1 − t)∂M. Here

∇h_{(1−t)M}(u) = ν^{−1}(u),

and ν : (1−t)∂M → S^{n−1} is the spherical image map (see [Ga2], pages 22-26, or [Sc2], pages 103-111). Since the Gaussian curvatures of M and B_{2}^{n} are positive everywhere, one can use the arguments which are similar to those in ([Sc2], pages 106-111), to show that the map u ∈ S^{n−1} → ∇h_{M}(u) is a C^{∞} diffeomorphism. Hence, the map u ∈ S^{n−1} → g_{t}(u) := (1 − t)∇h_{M}(u) + tu is also a C^{∞} diffeomorphism.

To show that t → ℜ^{−1}ρ_{K(t)}(ξ) is continuous, we pick any t ∈ [0, 1] and take any sequence {t_{m}}_{m=1}^{∞} of points from [0, 1] converging to t. The map

u ∈ S^{n−1} → f_{t}(u) := g_{t}(u)/|g_{t}(u)|

is a C^{∞} diffeomorphism for any t ∈ [0, 1], and f_{t_{m}} → f_{t} in C^{∞}(S^{n−1}). Hence, f_{t_{m}}^{−1} → f_{t}^{−1} in C^{∞}(S^{n−1}). Now, g_{t}(f_{t}^{−1}(ξ)) ∈ ∂K(t) implies ρ_{K(t)}(ξ) = |g_{t}(f_{t}^{−1}(ξ))|, and ρ_{K(t_{m})} converges to ρ_{K(t)} in C^{∞}(S^{n−1}). Since ℜ is a continuous bijection of C^{∞}(S^{n−1}) to itself, ([Ga2], page 382), the lemma is proved. □

Lemma 3.4. Let n ≥ 5. For any point ξ_{0} ∈ S^{n−1} there exists K̃ ∈ C_{+}^{∞} such that ℜ^{−1}ρK̃(ξ) is strictly positive for all ξ ≠ ±ξ_{0}, and ℜ^{−1}ρK̃(±ξ_{0}) = 0.

Proof. Fix n ≥ 5. Then there exists M ∈ C_{+}^{∞} such that ℜ^{−1}ρ_{M}(ξ) is sign-changing (see [K], Lemma 4.10 where an example of such body is constructed).

For t ∈ [0, 1], consider the Minkowski sum K(t) = tB_{2}^{n}+(1−t)M. Then ℜ^{−1}ρ_{K(0)}(ξ) is sign-changing and there exists Λ′ ⊂ S^{n−1} such that ℜ^{−1}ρ_{K(0)}(ξ) < 0, ∀ξ ∈ Λ′. On the other hand, ℜ^{−1}ρ_{K(1)}(ξ) > 0, ∀ξ ∈ S^{n−1}. By the previous lemma the map t → ℜ^{−1}ρ_{K(t)}(ξ) is continuous, and there is t_{0} ∈ [0, 1] such that

ℜ^{−1}ρ_{K(t_{0})}(ξ) ≥ 0, ∀ξ ∈ S^{n−1}  and  ℜ^{−1}ρ_{K(t_{0})}(ξ) = 0, ∀ξ ∈ Λ ⊂ S^{n−1},

for some Λ ≠ ∅. Fix any ξ_{0} ∈ Λ. Consider an even C^{∞} smooth function g on S^{n−1} such that

g(x) > 0, ∀x ≠ ±ξ_{0} and g(±ξ_{0}) = 0.

For ε > 0 define a body K̃ (depending on ξ_{0}):

ℜ^{−1}ρK̃(ξ) = ℜ^{−1}ρ_{K(t_{0})}(ξ) + εg(ξ).

Note that ℜ^{−1}ρK̃(ξ) is strictly positive for all ξ ≠ ±ξ_{0}, and ℜ^{−1}ρK̃(±ξ_{0}) = 0. We get

ρK̃(x) = ρ_{K(t_{0})}(x) + εℜg(x).

Since ℜg is a C^{∞} function, and K(t_{0}) ∈ C_{+}^{∞}, we may choose ε small enough so that K̃ ∈ C_{+}^{∞}. Using the rotation argument, we can take ξ_{0} to be arbitrary. □

Theorem 3.5. Let n ≥ 5 be odd. There exists ε > 0 and a convex symmetric body K that is not an intersection body, but nevertheless ∀x ∈ S^{n−1} there exists an intersection body L_{x} such that ρ_{K} = ρ_{L_{x}} on E_{ε}(x).

Proof. We define a convex body K and a family of convex bodies {L_{x}}_{x∈S^{n−1}} using K̃ and functions f_{x,ξ_{0}} from Lemma 3.1. We fix some small ε satisfying the requirements of Lemma 3.1 and we may assume that c = 2ℜ^{−1}1 (see Remark 3.2). Then, define K = K_{δ,ξ_{0}} via ρ_{K} = ρK̃ − δ, where for the moment δ > 0 is assumed to be so small that K ∈ C_{+}^{∞} and ℜ^{−1}ρ_{K} is strictly positive outside U_{ε}(ξ_{0}). Note that ℜ^{−1}ρ_{K}(ξ_{0}) < 0 and thus K is not an intersection body.

Now we define a family of convex bodies {L_{x}}_{x∈S^{n−1}}. Since K̃ ∈ C_{+}^{∞}, we take δ so small that ρ_{L_{x}} := ρK̃ − δ + δf_{x,ξ_{0}} > 0 on S^{n−1} and L_{x} is convex. Observe that ρ_{L_{x}} = ρ_{K} on E_{ε}(x) for any x ∈ S^{n−1}.

We can assume that δ is so small that

ℜ^{−1}ρ_{L_{x}} = ℜ^{−1}ρK̃ − δℜ^{−1}1 + δℜ^{−1}f_{x,ξ_{0}} > 0

on S^{n−1} \ U_{ε}(ξ_{0}), since ℜ^{−1}ρK̃ > 0 on S^{n−1} \ U_{ε}(ξ_{0}).

To show that bodies L_{x} are intersection bodies ∀x ∈ S^{n−1}, it is enough to prove that ℜ^{−1}ρ_{L_{x}} > 0 on U_{ε}(ξ_{0}). By Remark 3.2, min_{x∈S^{n−1}} ℜ^{−1}f_{x,ξ_{0}} ≥ 2ℜ^{−1}1 on U_{ε}(ξ_{0}), hence

ℜ^{−1}ρ_{L_{x}} = ℜ^{−1}ρK̃ − δℜ^{−1}1 + δℜ^{−1}f_{x,ξ_{0}} ≥ δℜ^{−1}1 > 0

on U_{ε}(ξ_{0}). Moreover, δ > 0 can be chosen independently of x since the set of functions {f_{x,ξ}}_{x,ξ∈S^{n−1}} in Lemma 3.1 is finite.

□

4. There is no local equatorial characterization of zonoids in odd dimensions.

The proofs in this section are very similar (in fact, almost identical) to the ones in the previous section.

Lemma 4.1. Let n ≥ 3 be odd. Then there exists an ε > 0 and an absolute constant c > 0 such that for any x, ξ ∈ S^{n−1}, there exists an even function f_{x,ξ} satisfying f_{x,ξ} = 0 on E_{ε}(x), and Cos^{−1}f_{x,ξ} ≥ c on U_{ε}(ξ).

Proof. The proof follows the same lines as that of Lemma 3.1. One has to change the Spherical Radon transform to the Cosine transform, put support functions instead of radial functions and thus, use section functions of polar bodies together with (2),

(4) and (8).

□

Remark 4.2. Note that dilating M and Q (and thus functions f_{x,ξ}) we may assume that c is as large as we want. For technical reasons, we take c = 2Cos^{−1}1. Moreover, we can assume that the set of functions {f_{x,ξ}}_{x,ξ∈S^{n−1}} in the lemma is finite.

Lemma 4.3. Let n ≥ 3. For any point ξ_{0} ∈ S^{n−1} there exists a zonoid K̃ ∈ C_{+}^{∞} such that Cos^{−1}hK̃(ξ) is strictly positive for all ξ ≠ ±ξ_{0}, and Cos^{−1}hK̃(±ξ_{0}) = 0.

Proof. Fix n ≥ 3. Then there exists M ∈ C_{+}^{∞} such that Cos^{−1}h_{M} is sign-changing (see [K], page 161, the Fourier Analytic solution of Shephard problem for a construction of a C_{+}^{∞} non-zonoid body).

For t ∈ [0, 1] consider the Minkowski sum K(t) = tB_{2}^{n} + (1 − t)M. Then h_{K(t)} = th_{B_{2}^{n}} + (1 − t)h_{M} is a C^{∞}-function, Cos^{−1}h_{K(0)}(ξ) is sign-changing and there exists Λ′ ⊂ S^{n−1} such that Cos^{−1}h_{K(0)}(ξ) < 0, ∀ξ ∈ Λ′. On the other hand, Cos^{−1}h_{K(1)}(ξ) > 0, ∀ξ ∈ S^{n−1}. The map t → Cos^{−1}h_{K(t)} is continuous, since Cos is a continuous bijection of C^{∞}(S^{n−1}) into itself, ([Ga2], page 381). Hence, there is t_{0} ∈ [0, 1] such that

Cos^{−1}h_{K(t_{0})} ≥ 0,  and  Cos^{−1}h_{K(t_{0})}(ξ) = 0, ∀ξ ∈ Λ ⊂ S^{n−1}

and some Λ ≠ ∅. Fix any ξ_{0} ∈ Λ. Consider an even C^{∞} smooth function g on S^{n−1} such that

g(x) > 0, ∀x ≠ ±ξ_{0} and g(±ξ_{0}) = 0.

For ε > 0 define a body K̃:

Cos^{−1}hK̃(ξ) = Cos^{−1}h_{K(t_{0})}(ξ) + εg(ξ).

Note that Cos^{−1}hK̃(ξ) is strictly positive for all ξ ≠ ±ξ_{0}, and Cos^{−1}hK̃(±ξ_{0}) = 0. Moreover,

hK̃ = h_{K(t_{0})} + εCosg.

Since Cosg is a continuous function and K(t_{0}) ∈ C_{+}^{∞}, we may choose ε small enough so that K̃ ∈ C_{+}^{∞}. Using the rotation argument, we can take ξ_{0} to be arbitrary. □

Theorem 4.4. Let n ≥ 3 be odd. There exists ε > 0 and a convex body K that is not a zonoid, but nevertheless ∀x ∈ S^{n−1} there exists a zonoid L_{x} such that h_{K} = h_{L_{x}} on E_{ε}(x).

Proof. We define a convex body K and a family of convex bodies {L_{x}}_{x∈S^{n−1}} using the zonoid K̃ and functions f_{x,ξ_{0}} from Lemma 4.1. We fix some small ε satisfying the requirements of Lemma 4.1 with c = 2Cos^{−1}1 (see Remark 4.2). Then, define K = K_{δ,ξ_{0}} via h_{K} = hK̃−δ, where for the moment δ > 0 is assumed to be so small that K ∈ C_{+}^{∞} and Cos^{−1}h_{K} is strictly positive outside U_{ε}(ξ_{0}). Note that Cos^{−1}h_{K}(ξ_{0}) < 0 and thus K is not a zonoid.

Now we define a family of convex bodies {L_{x}}_{x∈S^{n−1}}. Since K̃ ∈ C_{+}^{∞}, we take δ so small that h_{L_{x}} := hK̃ − δ + δf_{x,ξ_{0}} > 0 on S^{n−1} and L_{x} is convex. Observe that h_{L_{x}} = h_{K} on E_{ε}(x) for any x ∈ S^{n−1}.

We can assume that δ is so small that

Cos^{−1}h_{L_{x}} = Cos^{−1}hK̃ − δCos^{−1}1 + δCos^{−1}f_{x,ξ_{0}} > 0

on S^{n−1} \ U_{ε}(ξ_{0}), since Cos^{−1}hK̃ > 0 on S^{n−1} \ U_{ε}(ξ_{0}).

To show that bodies L_{x} are zonoids ∀x ∈ S^{n−1}, it is enough to prove that Cos^{−1}h_{L_{x}} > 0 on U_{ε}(ξ_{0}). By Remark 4.2, min_{x∈S^{n−1}} Cos^{−1}f_{x,ξ_{0}} > 2Cos^{−1}1 on U_{ε}(ξ_{0}), hence

Cos^{−1}h_{L_{x}} = Cos^{−1}hK̃ − δCos^{−1}1 + δCos^{−1}f_{x,ξ_{0}} ≥ δCos^{−1}1 > 0

on U_{ε}(ξ_{0}), and the result follows. □

5. There is a local equatorial characterization of intersection bodies and zonoids in even dimensions.

We consider at first intersection bodies. The proof of the following lemma is obtained by a straightforward repetition of the argument from ([K], page 60), and we omit the details.

Lemma 5.1. Let g(x) be an even homogeneous function of degree −1 such that g(x) is nonnegative and infinitely smooth on S^{n−1}. Then

ĝ(ξ) = (−1)^{(n−2)/2}πA_{g,ξ}^{(n−2)}(0).

where

A_{g,ξ}(z) =  ∫_{{y∈ℝ^{n}: y ·ξ=z}} χ_{[0,1]}(1/g(y))dy,  ξ ∈ S^{n−1}.

Theorem 5.2. Let n be even and let K ⊂ ℝ^{n} be an origin-symmetric convex body. Assume that for any great sphere ξ^{⊥}∩S^{n−1}, there exists an intersection body L_{ξ} and a neighborhood E_{ε(ξ)}(ξ) of ξ^{⊥}∩S^{n−1} such that the radial functions of K and L_{ξ} coincide at all points of E_{ε(ξ)}(ξ); then K is an intersection body.

Proof. If K and L_{ξ} are infinitely smooth, then it is enough to observe that ρ_{K}(u) = ρ_{L_{ξ}}(u) ∀u ∈ E_{ε(ξ)}(ξ) implies A_{K,ξ}(t) = A_{L_{ξ},ξ}(t) for sufficiently small t and apply (5).

Consider the general case. It was proved by Koldobsky that an original-symmetric body K is an intersection body if and only if ρ_{K} represents a positive definite distribution (see, for example, Theorem 4.1 in [K]). Thus, it is enough to show that

⟨ρ̂_{K}, φ⟩ ≥ 0, for all nonnegative test functions φ ∈ ℝ^{n}.

Using the definition of the Fourier Transform of distributions (see Section 2.5, [K]) and passing to the polar coordinates, we get

⟨ρ̂_{K}, φ⟩ = ⟨ρ_{K}, φ̂⟩ = ∫_{ℝ^{n}} ρ_{K}(x)φ̂(x)dx = ∫_{S^{n−1}} ρ_{K}(θ) ∫_{0}^{∞} r^{n−2}φ̂(rθ)dr dθ.

Observe that the function α(x) := ∫_{0}^{∞} r^{n−2}φ̂(rx)dr, x ∈ R^{n−1} \ {0} is homogeneous of degree −n + 1 and infinitely smooth. We may apply equality 4.3, page 72 in [K] together with Lemma 3.7 from[K] to claim that there exists an infinitely smooth nonnegative homogenous function g on S^{n−1}, such that ĝ(θ) = α(θ) and thus

∫_{S^{n−1}} ρ_{K}(θ) ∫_{0}^{∞} r^{n−2}φ̂(rθ)dr dθ = ∫_{S^{n−1}} ρ_{K}(θ)ĝ(θ)dθ.

Using a partition of unity on S^{n−1} together with homogenuity of g, we can write

g = ∑_{j=1}^{m} g_{j}, where supp g_{j}|_{S^{n−1}} ⊂ U_{ε_{j}}(ξ_{j}) are small enough. By the previous lemma,

supp g_{j}|_{S^{n−1}} ⊂ U_{ε_{j}}(ξ_{j}) implies supp ĝ_{j}|_{S^{n−1}} ⊂ E_{ε_{j}}(ξ_{j}). Hence,

⟨ρ̂_{K}, φ⟩ = ∑_{j=1}^{m} ∫_{S^{n−1}} ρ_{K}(θ)ĝ_{j}(θ)dθ = ∑_{j=1}^{m} ∫_{E_{ε_{j}} (ξ_{j})} ρ_{K}(θ)ĝ_{j}(θ)dθ =

∑_{j=1}^{m} ∫_{E_{ε_{j}} (ξ_{j})} ρ_{L_{ξ_{j}}} (θ)ĝ_{j}(θ)dθ = ∑_{j=1}^{m} ∫_{S^{n−1}} ρ_{L_{ξ_{j}}} (θ)ĝ_{j}(θ)dθ = ∑_{j=1}^{m}⟨ρ̂_{L_{ξ_{j}}}, φ_{j}⟩ ≥ 0,

where the last equality, again, follows from 4.3, page 72 in [K] and Lemma 3.7 from[K], which guarantee that φ_{j} ≥ 0. □

The following result was obtained independently by G. Panina [Pan] and P. Goodey and W. Weil [GW]. Its proof could be also obtained by the arguments similar to those in the previous proof, and we omit it.

Theorem 5.3. Let n be even and let K ⊂ ℝ^{n} be an origin-symmetric convex body. Assume that for any great sphere ξ^{⊥} ∩ S^{n−1}, there exists a zonoid Z_{ξ} and a neighborhood E_{ε(ξ)}(ξ) of ξ^{⊥} ∩ S^{n−1} such that the boundaries of K and Z_{ξ} coincide at all points where the exterior unit vector belong to E_{ε(ξ)}(ξ); then K is a zonoid.

6. There is no local characterization of intersection bodies.

In this section we prove the analog of the result of W. Weil [W] for zonoids. Our proof is different from the one of W. Weil. We show that, given x, ξ ∈ S^{n−1}, one can construct a function f which is zero around x, but such that the inverse spherical Radon transform of f is positive around ξ. For convenience of the reader we split the proof of this auxiliary result (see Lemma 6.4) into four statements. We will use the following notation

ℑ_{ε,x} = {f ∈ C^{∞}(S^{n−1}) : f = 0 on U_{ε}(x)}, 0 < ε < 1.

Lemma 6.1. Let n ≥ 3, and let ξ, x ∈ S^{n−1} be two orthogonal vectors. Assume that any f ∈ ℑ_{1/4,x} satisfies ℜ^{−1}f(ξ) = 0. Then for any pair of orthogonal vectors u, v ∈ S^{n−1} we have f ∈ ℑ_{1/4,u} implies ℜ^{−1}f(v) = 0.

Proof. For any two pairs of orthogonal unit vectors (ξ, x), (u, v) there exists a rotation ρ ∈ SO(n) satisfying u = ρ(x), v = ρ(ξ). Since ℜ^{−1} commutes with rotations, the result follows. □

Lemma 6.2. Let n ≥ 3, and let ξ ∈ x^{⊥}. Assume that any f ∈ ℑ_{1/4,x} satisfies ℜ^{−1}f(ξ) = 0. Then ℜ^{−1}(ℑ_{1/2,x}) ⊂ ℑ_{1/4,ξ}.

Proof. Take any u ∈ U_{1/4}(ξ). Let ρ ∈ SO(n), ρ(ξ) = u, where ξ is rotated into u inside U_{1/4}(ξ) in the plane containing ξ, u and the origin. Then ρ(x) ∈ U_{1/4}(x), and ℑ_{1/2,x} ⊂ ℑ_{1/4,ρ(x)}. Moreover, ℜ^{−1}f(u) = 0 since ℜ^{−1} commutes with rotations. The point u was chosen arbitrarily in U_{1/4}(ξ), hence ℜ^{−1}(ℑ_{1/2,x}) ⊂ ℑ_{1/4,ξ}. □

Lemma 6.3. Let n ≥ 3, and let ξ ∈ x^{⊥}. Then there exists a function f = f_{x,ξ} on S^{n−1} satisfying f_{x,ξ} = 0 on U_{1/4}(x), but ℜ^{−1}f_{x,ξ}(ξ) ≠ 0.

Proof. Assume the contrary. Then ℜ^{−1}(ℑ_{1/2,x}) ⊂ ℑ_{1/4,ξ} by Lemma 6.2. Take any vector y ∈ S^{n−1}, and find a vector q ∈ x^{⊥} ∩ y^{⊥}. Let ρ ∈ SO(n) be such that ρ(x) = x, ρ(ξ) = q. Observe that f ∈ ℑ_{ϵ,x} implies f(ρ(·)) ∈ ℑ_{ϵ,x}. Since ℜ^{−1} commutes with rotations, ℜ^{−1}(ℑ_{1/2,x}) ⊂ ℑ_{1/4,ξ} yields ℜ^{−1}(ℑ_{1/2,x}) ⊂ ℑ_{1/4,q}. Take two pairs of orthogonal vectors (x, q) and (q, y). By Lemma 6.1, we have ℜ^{−2}f(y) = 0. Thus, ℜ^{−2}f ≡ 0, a contradiction. □

Lemma 6.4. Let n ≥ 3. Then there exists an ε > 0 and an absolute constant c > 0 such that for any x, ξ ∈ S^{n−1}, there exists an even function f_{x,ξ} satisfying f_{x,ξ} = 0 on U_{ε}(x), and ℜ^{−1}f_{x,ξ} ≥ c on U_{ε}(ξ).

Proof. We fix points x and ξ, and provide an ε > 0, and c > 0 depending on x, ξ such that there is a function f_{x,ξ} satisfying f_{x,ξ} = 0 on U_{ε}(x), and ℜ^{−1}f_{x,ξ} ≥ c > 0 on U_{ε}(ξ). Then we use the compactness argument to prove the statement of the lemma.

Let ξ ∉ x^{⊥}. Then there exists an ε > 0, such that ξ ∉ E_{ε}(x). For any function g the values of ℜg on U_{ε}(x) depend only on the values of g on E_{ε}(x). Hence, we may consider an even C^{∞}-function g such that g(±ξ) > 0 and g(ν) = 0, for ν ∈ E_{ε}(x) and define f_{x,ξ} = ℜg(x).

Let ξ ∈ x^{⊥}. Then, the previous lemma implies the existence of ε = ε(x, ξ) = 1/8, and a function f = f_{x,ξ} on S^{n−1} satisfying f_{x,ξ} = 0 on U_{ε}(x), but ℜ^{−1}f_{x,ξ}(ξ) > 0 (change the sign of f_{x,ξ} if necessary).

Thus, we proved that for any x and ξ, there is ε′ = ε′(x, ξ) > 0 and there is a function f_{x,ξ} such that f_{x,ξ} = 0 on U_{ε′}(x), but ℜ^{−1}f_{x,ξ}(±ξ) ≥ c′, c′ = c′(x, ξ) > 0. From the continuity of the function ℜ^{−1}f_{x,ξ} we get that ℜ^{−1}f_{x,ξ} ≥ c, c = c(x, ξ) > 0 on U_{ε′′}(ξ), for some ε′′ > 0. Take ε̃ = ε̃(x, ξ) = min(ε′, ε′′). We show that for any x and ξ, there is ε̃ = ε̃(x, ξ) > 0 and there is a function f_{x,ξ} such that f_{x,ξ} = 0 on Uε̃(x), but ℜ^{−1}f_{x,ξ} ≥ c on Uε̃(ξ), c = c(x, ξ) > 0.

Now we use the compactness argument to prove that we can choose an ε and c independent of x and ξ. We choose a finite set of {x_{i}, ξ_{i}}_{i=1}^{m} such that {U_{ε̃_{i}/2}(x_{i}) × U_{ε̃_{i}/2}(ξ_{i})}_{i=1}^{m} covers S^{n−1} × S^{n−1}. We take

ε = \frac{1}{2} min_{1≤i≤m} ε̃_{i} and c = min_{1≤i≤m} c(x_{i}, ξ_{i}).

Then for any (x, ξ) there is a (x_{i}, ξ_{i}) such that

U_{ε}(x) × U_{ε}(ξ) ⊂ Uε̃_{i}(x_{i}) × Uε̃_{i}(ξ_{i}),

and we may define f_{x,ξ} = f_{x_{i},ξ_{i}}.

□

Theorem 6.5. Let n ≥ 5. There exists a convex body K that is not an intersection body, such that ∀x ∈ S^{n−1} there exists an ε(x) and an intersection body L_{x} such that ρ_{K} = ρ_{L_{x}} on U_{ε(x)}(x).

Proof. Repeat the proof of Theorem 1. □

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Fedor Nazarov, Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

  E-mail address: fedja@math.msu.edu

Dmitry Ryabogin, Department of Mathematics, Kansas State University, Manhattan, KS, 66506-2602, USA

  E-mail address: ryabs@math.ksu.edu

Artem Zvavitch, Department of Mathematics, Kent State University, Kent, OH, 44242, USA

  E-mail address: zvavitch@math.kent.edu