January 13, 2011

A PROBLEM OF KLEE ON INNER SECTION FUNCTIONS OF CONVEX BODIES

R. J. GARDNER, D. RYABOGIN, V. YASKIN, AND A. ZVAVITCH

Abstract. In 1969, Vic Klee asked whether a convex body is uniquely determined (up to translation and reflection in the origin) by its inner section function, the function giving for each direction the maximal area of sections of the body by hyperplanes orthogonal to that direction. We answer this question in the negative by constructing two infinitely smooth convex bodies of revolution about the x_{n}-axis in ℝ^{n}, n ≥ 3, one origin symmetric and the other not centrally symmetric, with the same inner section function. Moreover, the pair of bodies can be arbitrarily close to the unit ball.

1. Introduction

Let K be a convex body in ℝ^{n}. The inner section function m_{K} is defined by

m_{K}(u) = max_{t∈ℝ}V (K ∩ (u^{⊥} + tu)),

for u ∈ S^{n−1}. Here u^{⊥} denotes the hyperplane through the origin orthogonal to u and V denotes volume, that is, the k-dimensional Hausdorff measure of a k-dimensional body. (See Section 2 for other notation and definitions.) Thus the inner section function simply gives, for each direction, the maximal area of cross-sections orthogonal to that direction.

Interest in the inner section function goes back at least to the 1926 paper [1] of Bonnesen. It has been called the inner (n − 1)-quermass, particularly by authors who prefer to use the term “outer (n−1)-quermass” for the brightness function, the function giving the areas of the orthogonal projections of a body onto hyperplanes. Values of the inner section function have also been called HA-measurements, a term devolving from the de Haas-van Alphen effect in the study of Fermi surfaces of metals. The following explanation is taken from [6, Note 8.12]. The Fermi surface of a metal bounds a body formed, in velocity space, by velocity states occupied at absolute zero by valence electrons of the metal. The Pauli exclusion principle allows no more than two electrons (with opposite spins) to possess the same velocity (i.e., speed and direction), so electrons can only move into unoccupied states lying outside the Fermi surface. The more electrons there are near the Fermi surface, the larger the number that can increase their energy when the metal is heated and the larger the number whose spins can be aligned with a magnetic field. In this way the Fermi surface relates to the specific heat and magnetic properties of the metal, and the concept also provides an explanation of conductivity and ductility, for example. The body bounded by a Fermi surface is not generally convex. However, the definition of the inner section function above extends naturally to any bounded Borel set and for the body bounded by a Fermi surface it may actually be measured by means of the de Haas–van Alphen effect, i.e., magnetism induced in the metal by a strong magnetic field at a low temperature.

1991 Mathematics Subject Classification. Primary: 52A20, 52A40; secondary: 52A38.

Key words and phrases. convex body, intersection body, cross-section body, inner section function, HAmeasurement, geometric tomography.

First author supported in part by U.S. National Science Foundation Grant DMS-0603307. Second and fourth authors supported in part by U.S. National Science Foundation Grant DMS-0652684. Third author supported in part by NSERC. This research arose from discussions at the workshop on Mahler’s Conjecture and Duality in Convex Geometry at the American Institute of Mathematics, August 9–13, 2010.

In two influential articles published about 40 years ago, Klee [10], [11] asked whether a convex body in ℝ^{n}, n ≥ 3, is uniquely determined, up to translation and reflection in the origin, by its inner section function. The problem also appears in the books [2, p. 24] and [6, Problem 8.8(i)]. It has long been known that planar convex bodies whose inner section functions are constant are precisely the planar convex bodies of constant width. This explains the restriction n ≥ 3 in Klee’s problem, since when n = 2, any convex body of constant width that is not a disk provides a counterexample.

The main purpose of the present paper is to answer Klee’s question negatively. We do this by constructing two convex bodies in ℝ^{n}, n ≥ 3, each a body of revolution about the x_{n}-axis, such that one body is origin symmetric and the other is not centrally symmetric, while both have the same inner section function.

Klee’s problem is now seen as belonging to a fairly extensive literature around the important concept of an intersection body. Given a star body L in ℝ^{n}, i.e., a set star-shaped with respect to the origin whose radial function ρ_{L} is continuous, the intersection body IL of L is defined by

ρ_{IL}(u) = V (L ∩ u^{⊥}),

for u ∈ S^{n−1}. Clearly, IL is an origin-symmetric star body. Moreover, it is known that an origin-symmetric star body L is uniquely determined by the values V (L ∩ u^{⊥}), u ∈ S^{n−1}, and hence by its intersection body IL. This was proved by Funk [3] for convex bodies in ℝ^{3}; the general result is called Funk’s section theorem in [6, Corollary 7.2.7], though it was not stated explicitly in this generality until the work of Lifshitz and Pogorelov [13] on Fermi surfaces. Intersection bodies were introduced by Lutwak [14] and turned out to be the key to solving the Busemann-Petty problem (see [4], [7], and [17], as well as [6, Section 8.2] and [12, Chapter 5]).

For an origin-symmetric convex body K in ℝ^{n}, the Brunn-Minkowski inequality (see [5]) implies that m_{K}(u) = V (K ∩ u^{⊥}) for each u ∈ S^{n−1}. It follows immediately that Klee’s problem has an affirmative answer if set entirely within the class of origin-symmetric convex bodies. It is also known that in this case IK is itself an origin-symmetric convex body, by Busemann’s theorem; see, for example, [6, Theorem 8.1.10].

Now suppose that K is an arbitrary convex body in ℝ^{n}. The cross-section body CK of K is defined by

ρ_{CK}(u) = m_{K}(u),

for u ∈ S^{n−1}. Cross-section bodies, introduced by H. Martini, are the subject of [6, Section 8.3 and Note 8.12]. As we have seen, when K is origin symmetric, m_{K}(u) = V (K ∩ u^{⊥}) for each u ∈ S^{n−1}, and hence CK = IK is also an origin-symmetric convex body.

With this background, Klee’s question can be rephrased as asking whether a convex body in ℝ^{n}, n ≥ 3, is uniquely determined, up to translation and reflection in the origin, by its cross-section body. In these terms, our solution involves constructing a non-centrally-symmetric convex body K whose cross-section body CK is the (necessarily origin-symmetric and convex) intersection body IL of some origin-symmetric convex body L. Thus K and L are not equal, up to translation and reflection in the origin, while CK = IL = CL.

The paper is organized as follows. After the preliminary Section 2, the main Section 3 states our result and proves it via a succession of lemmas. The paper ends with the short Section 4 of concluding remarks.

2. Definitions, notation, and preliminaries

As usual, S^{n−1} denotes the unit sphere and o the origin in Euclidean n-space ℝ^{n}. The unit ball in ℝ^{n} will be denoted by B^{n}. The standard orthonormal basis for ℝ^{n} will be {e_{1}, …, e_{n}}. The inner product of vectors x and y is denoted by ⟨x, y⟩ and the Euclidean norm of x by |x|.

If X is a set, we denote by ∂X and int X the boundary and interior of X, respectively. The set −X is the reflection of X in the origin. A set X is origin symmetric if X = −X

and centrally symmetric if it is a translate of an origin-symmetric set.

A body is a compact set equal to the closure of its interior. If K is a k-dimensional body in ℝ^{n}, then V (K) is its volume H^{k}(K), where H^{k} is k-dimensional Hausdorff measure in ℝ^{n}, k = 1, …, n. The notation dz will always mean dH^{k}(z) for the appropriate k = 1, …, n. We follow Schneider [16] by writing κ_{n} and ω_{n} = nκ_{n} for the volume and surface area of the unit ball in ℝ^{n}, respectively.

A convex body is a compact convex set with nonempty interior.

If L is a body containing the origin in its interior and star-shaped with respect to the origin, its radial function ρ_{L} is defined by

ρ_{L}(x) = max{c ∈ ℝ : cx ∈ L},

for x ∈ ℝ^{n} \ {o}. Note that ρ_{L}(x) = 1 if and only if x ∈ ∂L. For u ∈ S^{n−1}, ρ_{L}(u) gives the distance from the origin to ∂L in the direction u. Moreover, ρ_{L} is homogeneous of degree −1, i.e., ρ(tx) = ρ(x)/t for t > 0. The homogeneity means that we can often regard ρ_{L} as a function on S^{n−1}.

For the purposes of this paper, a star body is a body whose radial function is continuous on S^{n−1}. (The reader should be aware that other definitions are often used.) When considering star bodies of revolution about the x_{n}-axis in ℝ^{n}, we can regard the radial function as a function of the spherical polar coordinate angle ϕ with the positive x_{n}-axis.

Let K be a convex body and let u ∈ S^{n−1}. The parallel section function A_{K,u}(t) is defined by

(1)  A_{K,u}(t) = V (K ∩ (u^{⊥} + tu)),

for t ∈ ℝ. See [12, Lemma 2.4 and Section 3.3] for recent results concerning this function. With this notation, the inner section function of K is

m_{K}(u) = max_{t∈ℝ}A_{K,u}(t),

for u ∈ S^{n−1}.

In the literature, it is often ignored that the parallel section function A_{K,u} is just the (n − 1)-dimensional X-ray of K in the direction u. See [6, Chapter 2]. Thus this function has a prominent role in Geometric Tomography, and indeed Klee’s problem was a significant open question in this subject.

As usual, C(X) and C^{k}(X), 1 ≤ k ≤ ∞, denote the classes of continuous and k times continuously differentiable functions on a subset X of ℝ^{n}, and C_{e}(X) and C_{e}^{k}(X) will signify the even functions in these classes.

We denote by R the spherical Radon transform, defined by

(Rf)(u) = ∫_{S^{n−1}∩u^{⊥}} f(v) dv,

for bounded Borel functions f on S^{n−1}. It is known (see [6, Theorem C.2.4]) that R is injective on the class of even functions on S^{n−1}, and this fact, applied to (n − 1)th powers of radial functions of star bodies, immediately yields Funk’s section theorem mentioned in the Introduction. As a historical aside, we remark that this injectivity property of R was proved in 1904 by Minkowski (see [6, p. 430]), who used it to show that a convex body in ℝ^{3} of constant girth also has constant width. However, Minkowski apparently did not notice the result of Funk (who proved it differently).

We adopt a standard definition of the Fourier transform f̂of a function f ∈ L_{1}(ℝ^{n}), namely

f̂(x) = ∫_{ℝ^{n}} f(y)e^{−i⟨x,y⟩} dy.

If f ∈ C(S^{n−1}) (or f ∈ C^{k}(S^{n−1})) and p ∈ ℝ, then f can be extended by setting

(2)  f(x) = |x|^{−n+p}f (\frac{x}{|x|}) ,

for x ∈ ℝ^{n} \{o}, and this extension is a homogeneous of degree −n+p function in C(ℝ^{n} \{o}) (or C^{k}(ℝ^{n} \ {o}), respectively). Note that throughout the paper, we shall use the same notation for a function on S^{n−1} and its homogeneous extension to ℝ^{n} \ {o}, since the distinction will be clear from the context. The Fourier transform of this extension of f, which we also denote by f̂, exists in the sense of distributions, as explained in [8, p. 389] or [12, Sections 2.5 and 3.3], for example. Where appropriate, statements in the sequel involving Fourier transforms of homogeneous extensions of functions on S^{n−1} are also to be considered in the sense of distributions.

Let D^{α}, where α = (α_{1}, …, α_{n}) is a multiindex of nonnegative integers, be the differential operator defined by

D^{α}f = \frac{∂^{|α|_{1}}f}{∂x_{1}^{α_{1}} ⋯ ∂x_{n}^{α_{n}}} ,

for f ∈ C^{|α|_{1}}(ℝ^{n}), where |α|_{1} = ∑_{j=1}^{n} α_{j}. Then

(3)  (D^{α}f)^{∧} = i^{|α|_{1}}x_{1}^{α_{1}} ⋯ x_{n}^{α_{n}} f̂

and

(4)  D^{α}f̂ = (−i)^{|α|_{1}} (x_{1}^{α_{1}} ⋯ x_{n}^{α_{n}}f)^{∧} .

These formulas also hold in the distributional sense; see, for example, [12, p. 35]. The relation

(5)  (△f)^{∧} = −|x|^{2}f̂,

where △ is the Laplacian operator in ℝ^{n}, follows directly from (3).

If f is even, then f̂ is also even. If f ∈ C_{e}(ℝ^{n} \ {o}) is homogeneous of degree −n + 1, then by [12, Lemma 3.7], f̂ is a homogeneous of degree −1 function in C_{e}(ℝ^{n} \ {o}) and

(6)  (Rf)(u) = \frac{1}{π} f̂(u),

for u ∈ S^{n−1}.

The spherical Radon transform is a continuous bijection from C_{e}^{∞}(S^{n−1}) to itself; see, for example, [6, Theorem C.2.5]. Let g ∈ C_{e}^{∞}(S^{n−1}), and extend g as in (2) (with f there replaced by g and p = n − 1) to a homogeneous of degree −1 function in C_{e}^{∞}(ℝ^{n} \ {o}). If f = ĝ, then, by [12, Lemma 3.16], f ∈ C_{e}^{∞}(ℝ^{n} \ {o}) is homogeneous of degree −n + 1. Since g is even, we have ˆ ĝ = (2π)^{n}g and from (6) it follows that

(7)  (R^{−1}g)(u) = \frac{1}{(2π)^{n}} (R^{−1}ˆĝ)(u) = \frac{1}{(2π)^{n}} (R^{−1}f̂)(u) = \frac{π}{(2π)^{n}} f(u) = \frac{π}{(2π)^{n}} ĝ(u),

for all u ∈ S^{n−1}.

A rotationally symmetric function f on S^{n−1} is one that can be defined via a function f(ϕ) of the vertical angle ϕ in spherical polar coordinates by setting f(u) = f(arccos u_{n}) = f(ϕ) for u = (u_{1}, …, u_{n}) ∈ S^{n−1}. Moreover, we will consider f as an even function on S^{1}, by first extending f to [−π, π] by letting f(−ϕ) = f(ϕ) and then regarding S^{1} as [−π, π] with its endpoints identified. Of course we are abusing notation by using the same letter for the function of u ∈ S^{n−1}, the function of ϕ ∈ [0, π], and the function of ϕ ∈ S^{1}, but will adopt this convenient practise throughout the paper.

3. Main result

Theorem 3.1. There exist convex bodies K and L in R^{n}, n ≥ 3, with ρ_{K}, ρ_{L} ∈ C^{∞}(S^{n−1}), each a body of revolution about the x_{n}-axis, such that K is not centrally symmetric, L is origin symmetric, and m_{K} = m_{L}.

The theorem will be proved in a series of lemmas.

Lemma 3.2. Let 0 < ε < 1 and let K_{ε} be the body of revolution about the x_{n}-axis in ℝ^{n} with radial function given in spherical polar coordinates by

(8)  ρ_{K_{ε}}(ϕ) = (1 + ε cos^{3} ϕ)^{−1/3} ,

where 0 ≤ ϕ ≤ π is the angle with the positive x_{n}-axis, or equivalently by

(9)  ρ_{K_{ε}}(x) = (|x|^{3} + εx_{n}^{3})^{−1/3},

where x = (x_{1}, …, x_{n}) ∈ ℝ^{n} \ {o}. Then ρ_{K_{ε}} ∈ C^{∞}(S^{n−1}), the body K_{ε} is not centrally symmetric, and there is a positive ε_{0} ≤ 1 such that K_{ε} is convex for all ε < ε_{0}.

Proof. It is clear from its definition via (9) that ρ_{K_{ε}} ∈ C^{∞}(S^{n−1}) provided ε < 1. To prove that K_{ε} is not centrally symmetric and is convex for sufficiently small ε, it suffices to show that this is true for the planar convex body J_{ε} obtained by intersecting K_{ε} with the x_{1}, x_{n}-plane. The radial function of J_{ε} is also given by (8), where we can regard ϕ as the polar coordinate angle in the x_{n}, x_{1}-plane (note that ρ_{J_{ε}}(−ϕ) = ρ_{J_{ε}}(ϕ)). It is well known that the curvature of a planar C^{2} curve given in polar coordinates by r = r(θ) is

(10)  \frac{2(r′)^{2} − rr′′ + r^{2}}{((r′)^{2} + r^{2})^{3/2}} .

Using this, we find that the curvature of J_{ε} is

\frac{(1 + ε cos^{3} ϕ + 2ε sin^{2} ϕ cos ϕ)(1 + ε cos^{3} ϕ)^{4/3}}{((1 + ε cos^{3} ϕ)^{2} + ε^{2} cos^{4} ϕ sin^{2} ϕ)^{3/2}} .

When ε is sufficiently small, the curvature is positive and hence J_{ε} is convex. Also, the curvatures when ϕ = 0 and ϕ = π are (1 + ε)^{−2/3} and (1 − ε)^{−2/3}, respectively. It follows that J_{ε} is not centrally symmetric, since a center of symmetry would have to lie on the x_{n}-axis, and ϕ = 0 and ϕ = π correspond to antipodal points in ∂J_{ε}. □

The body K in Theorem 3.1 will be K = K_{ε} for a suitable ε. From now on we assume that ε < ε_{0}. It can be seen numerically that one can take ε_{0} = 0.91, but note that in any case ε_{0} < 1 by the definition of K_{ε}. Since K_{ε} is a body of revolution about the x_{n}-axis, for any fixed t ∈ ℝ, the function A_{K_{ε},u}(t), u ∈ S^{n−1} defined by (1) is rotationally symmetric. Therefore we can write A_{K_{ε},u}(t) = A_{K_{ε}, ϕ}(t), where we can either consider ϕ ∈ [0, π] as the angle between u and e_{n} or as an element ϕ ∈ S^{1}, as explained at the end of Section 2.

Lemma 3.3. Let n ≥ 3 and let K_{ε} be as in Lemma 3.2. Then

(i) For each ϕ ∈ S^{1}, the parallel section function A_{K_{ε}, ϕ}(t) has a maximum at a unique point t = t_{ε}(ϕ).

(ii) There is a positive ε_{1}(n) ≤ ε_{0} such that

(11)  t_{ε}(ϕ) = T_{ε}(ϕ)ε,

where T_{ε}(ϕ) ∈ C^{∞}(S^{1}) for all 0 < ε < ε_{1}(n).

(iii) For k = 0, 1, …, there is a constant c_{1}(k, n) such that

(12)  |\frac{d^{k}T_{ε}(ϕ)}{dϕ^{k}} | ≤ c_{1}(k, n),

for all 0 < ε < ε_{1}(n) and ϕ ∈ S^{1}.

Proof. (i) The Brunn-Minkowski inequality (see [5] or [6, Section B.2]) implies that for each ϕ the function A_{K_{ε}, ϕ}(t)^{1/(n−1)} is concave and hence A_{K_{ε}, ϕ}(t) is unimodal, i.e., increases and then decreases with t. Therefore, if its maximum is not attained at a unique point, it is attained at all points in a closed interval [t_{1}, t_{2}], say, t_{1} ≠ t_{2}. Now the equality condition of the Brunn-Minkowski inequality implies that the sections K_{ε} ∩ (u^{⊥} + tu), t ∈ [t_{1}, t_{2}] are all homothetic and hence, by convexity, the union of these sections is a (possibly slanted) cylinder. Since K_{ε} is a body of revolution, it is easy to see that this is only possible when the cylinder is a right spherical cylinder with its axis along the x_{n}-axis. However, this implies that K_{ε} has vertical line segments in its boundary, which clearly contradicts its definition via (8).

(ii) Let u ∈ S^{n−1} and t ∈ ℝ be such that tu ∈ int K_{ε}. Note that for any v ∈ S^{n−1} ∩ u^{⊥}, we have ρ_{K_{ε}−tu}(v)v ∈ ∂(K_{ε} − tu) and hence x = tu + ρ_{K_{ε}−tu}(v)v ∈ ∂K_{ε}. Then ρ_{K_{ε}}(x) = 1 and hence (9) yields

(13)  1 = (t^{2} + ρ_{ε}^{2})^{3/2} + ε(t cos ϕ + ρ_{ε}v_{n})^{3},

where ϕ ∈ [0, π], v = (v_{1}, …, v_{n}), and where for brevity we write ρ_{ε} = ρ_{K_{ε}−tu}(v). If we set

(14)  e_{n} = u cos ϕ + w sin ϕ,

where w ∈ S^{n−1} ∩ u^{⊥}, and s = ⟨v, w⟩, then

v_{n} = ⟨v, e_{n}⟩ = ⟨v, w⟩ sin ϕ = s sin ϕ.

Then we can rewrite (13) as

(15)  1 = (t^{2} + ρ_{ε}^{2})^{3/2} + ε(t cos ϕ + ρ_{ε}s sin ϕ)^{3},

where ϕ ∈ [0, π] and −1 ≤ s ≤ 1. Thus the radial function of (K_{ε} − tu) ∩ u^{⊥} = K_{ε} ∩ (u^{⊥} + tu), as a function ρ_{ε} = ρ_{ε}(t, ϕ, s), is implicitly defined by (15). We can also regard ρ_{ε} as defined for ϕ ∈ S^{1} by setting

(16)  ρ_{ε}(t, ϕ, s) = ρ_{ε}(t, −ϕ, −s),

for ϕ ∈ [−π, 0].

We claim that there is a positive δ_{1} ≤ ε_{0} such that for 0 < ε < δ_{1}, we have |t_{ε}(ϕ)| < 1/4 and the function ρ_{ε} = ρ_{ε}(t, ϕ, s) implicitly defined by (15) is infinitely differentiable with respect to each variable on (−1/4, 1/4) × S^{1} × [−1, 1]. (At the endpoints s = −1 and s = 1 of [−1, 1] we take one-sided derivatives.) To see this, note first that by (8), K_{ε} → B^{n} in the Hausdorff metric as ε → 0+. Consequently, there is a positive δ_{0} ≤ ε_{0} such that for 0 < ε < δ_{0}, we have |t_{ε}(ϕ)| < 1/4 and 3/4 < ρ_{K_{ε}}(ϕ) < 5/4 for all ϕ ∈ S^{1}. Then 1/2 < ρ_{ε} = ρ_{K_{ε}−tu}(v) < 3/2 when |t| < 1/4 and 0 < ε < δ_{0}. Let F_{ε} : ℝ^{4} → ℝ be defined by

(17)  F_{ε}(x_{1}, x_{2}, x_{3}, y) = (x_{1}^{2} + y^{2})^{3/2} + ε(x_{1} cos x_{2} + yx_{3} sin x_{2})^{3} − 1.

Then we can choose 0 < δ_{1} ≤ δ_{0} small enough to ensure that if 0 < ε < δ_{1}, then

(18)  \frac{∂F_{ε}}{∂y} = 3(x_{1}^{2} + y^{2})^{1/2}y + 3ε(x_{1} cos x_{2} + yx_{3} sin x_{2})^{2}x_{3} sin x_{2} > 3/4 − 3(7/4)^{2}ε > 0,

for all (x_{1}, x_{2}, x_{3}, y) ∈ (−1/4, 1/4)×[−π, π]×[−1, 1]×(1/2, 3/2). Comparing (15) and (17) and noting that F_{ε} is periodic with period 2π in the variable x_{2}, we see that the implicit function theorem implies that for all 0 < ε < δ_{1}, there is a unique function ρ_{ε} = ρ_{ε}(t, ϕ, s) that satisfies (15) and is infinitely differentiable with respect to each variable on (−1/4, 1/4) × S^{1} × [−1, 1]. This proves the claim.

Differentiation of (15) with respect to t yields

(19)  \frac{∂ρ_{ε}}{∂t} = −\frac{(t^{2} + ρ_{ε}^{2})^{1/2}t + ε(t cos ϕ + ρ_{ε}s sin ϕ)^{2} cos ϕ}{(t^{2} + ρ_{ε}^{2})^{1/2}ρ_{ε} + ε(t cos ϕ + ρ_{ε}s sin ϕ)^{2}s sin ϕ}.

Now ρ_{ε}, s, and t are bounded above uniformly in ε for 0 < ε < δ_{0}. Also, since ρ_{ε} > 1/2 when |t| < 1/4, the presence of the term (t^{2} + ρ_{ε}^{2})^{1/2}ρ_{ε} in the previous denominator guarantees that this denominator is bounded below by a positive constant, uniformly in ε for 0 < ε < δ_{1}. (We can use δ_{1} here in view of (18).) Moreover, the same denominator appears in ∂ρ_{ε}/∂ϕ and ∂ρ_{ε}/∂s, and powers of it appear in all higher derivatives with respect to t, ϕ, and s. Therefore the partial derivatives of ρ_{ε} of any order are each bounded uniformly in ε for 0 < ε < δ_{1}.

Next, we observe that if α is the angle between v and the w defined by (14), then the set of points in S^{n−1} ∩ u^{⊥} having the same fixed angle α at o is an (n − 3)-sphere of radius sin α. Then s = sin α and since A_{K_{ε},ϕ}(t) = A_{K_{ε}−tu,ϕ}(0), we have

A_{K_{ε},ϕ}(t) = V ((K_{ε} − tu) ∩ u^{⊥}) = \frac{1}{n − 1} ∫_{S^{n−1}∩u^{⊥}} ρ_{ε}(t, ϕ, cos α)^{n−1} dv
= \frac{ω_{n−2}}{n − 1} ∫_{0}^{π} ρ_{ε}(t, ϕ, cos α)^{n−1} sin^{n−3} α dα
(20)  = \frac{ω_{n−2}}{n − 1} ∫_{−1}^{1} ρ_{ε}(t, ϕ, s)^{n−1} dμ(s),

where dμ(s) = (1 − s^{2})^{(n−4)/2}ds. (The geometry behind the substitution in the integral is depicted in [6, Figure 8.4, p. 314]; compare the proof of [6, Theorem C.2.9]. See also [9, Lemma 1.3.1(ii)].) Recalling that the integrand in (20) can be regarded as defined for ϕ ∈ S^{1} via (16), we see that we can regard A_{K_{ε},ϕ}(t) as an even function of ϕ ∈ S^{1} in the sense that A_{K_{ε},−ϕ}(t) = A_{K_{ε},ϕ}(t) for all ϕ ∈ S^{1}.

The restriction n ≥ 3 allows differentiation under the integral sign in integrals with respect to dμ(s) such as (20) for which the integrand is continuously differentiable with respect to its variables. We shall use this several times in the sequel without further comment.

At the unique point t = t_{ε}(ϕ) where A_{K_{ε},ϕ}(t) attains its maximum, we have, by differentiating (20) with respect to t and using (19), that

A′_{K_{ε},ϕ}(t) = ω_{n−2} ∫_{−1}^{1} ρ_{ε}(t, ϕ, s)^{n−2} \frac{∂ρ_{ε}}{∂t} (t, ϕ, s) dμ(s)
(21)  = −ω_{n−2} ∫_{−1}^{1} ρ_{ε}^{n−2} \frac{(t^{2} + ρ_{ε}^{2})^{1/2}t + ε(t cos ϕ + ρ_{ε}s sin ϕ)^{2} cos ϕ}{(t^{2} + ρ_{ε}^{2})^{1/2}ρ_{ε} + ε(t cos ϕ + ρ_{ε}s sin ϕ)^{2}s sin ϕ} dμ(s) = 0,

where ρ_{ε} = ρ_{ε}(t, ϕ, s).

When ε = 0, we have K_{ε} = B^{n}, and from (15), ρ_{0} = \sqrt{1 − t^{2}}. Then differentiation of (21) with ε = 0, with respect to t, yields

A′′_{K_{ε},ϕ}(t) = −ω_{n−2} ∫_{−1}^{1}(1 − t^{2})^{(n−3)/2} (1 − \frac{(n − 3)t^{2}}{1 − t^{2}} ) dμ(s) < 0

when n = 3 and when n ≥ 4 and |t| < 1/\sqrt{n − 2}. Since ρ_{ε} and all its derivatives are continuous with respect to ε, we have A′′_{K_{ε},ϕ}(t) < 0 for all sufficiently small ε > 0, |t| < 1/(2\sqrt{n − 2}), say, and ϕ ∈ S^{1}. Now for sufficiently small ε > 0, we have |t_{ε}(ϕ)| < 1/(2\sqrt{n − 2}), where t_{ε}(ϕ) was defined in (i). Hence, by the implicit function theorem, for sufficiently small ε > 0, the unique t with |t| < 1/(2\sqrt{n − 2}) that satisfies (21) is t = t_{ε}(ϕ), and moreover t_{ε}(ϕ) ∈ C^{∞}(S^{1}). Summarizing, we have shown that there is a positive ε_{1}(n) ≤ δ_{1} such that for all 0 < ε < ε_{1}(n), t = t_{ε}(ϕ) satisfies (21) and t_{ε}(ϕ) ∈ C^{∞}(S^{1}).

Let 0 < ε < ε_{1}(n). Rearranging (21), we obtain

(22)  t_{ε}(ϕ) = −\frac{∫_{−1}^{1} Φ(t_{ε}(ϕ), ϕ, s) dμ(s)}{∫_{−1}^{1} Ψ(t_{ε}(ϕ), ϕ, s) dμ(s)}ε = T_{ε}(ϕ)ε,

say, where

(23)  Φ = \frac{ρ_{ε}^{n−2} (t_{ε}(ϕ) cos ϕ + ρ_{ε}s sin ϕ)^{2} cos ϕ}{(t_{ε}(ϕ)^{2} + ρ_{ε}^{2})^{1/2}ρ_{ε} + ε(t_{ε}(ϕ) cos ϕ + ρ_{ε}s sin ϕ)^{2}s sin ϕ}

and

(24)  Ψ = \frac{ρ_{ε}^{n−2} (t_{ε}(ϕ)^{2} + ρ_{ε}^{2})^{1/2}}{(t_{ε}(ϕ)^{2} + ρ_{ε}^{2})^{1/2}ρ_{ε} + ε(t_{ε}(ϕ) cos ϕ + ρ_{ε}s sin ϕ)^{2}s sin ϕ}.

Here ρ_{ε} = ρ_{ε}(t_{ε}(ϕ), ϕ, s) and Φ and Ψ also involve t = t_{ε}(ϕ). Nevertheless, we have shown that t_{ε}(ϕ) = T_{ε}(ϕ)ε, say, where T_{ε}(ϕ) ∈ C^{∞}(S^{1}) because t_{ε}(ϕ) ∈ C^{∞}(S^{1}). This completes the proof of (ii).

(iii) Let 0 < ε < ε_{1}(n). In view of (22), (23), and (24) and the previously established bounds |t_{ε}(ϕ)| < 1/4 and 1/2 < ρ_{ε} < 3/2 that hold when 0 < ε < ε_{1}(n), there is a constant c_{1}(0, n) such that |T_{ε}(ϕ)| ≤ c_{1}(0, n) for all ϕ ∈ S^{1}. This proves (12) for k = 0.

To prove (12) for k = 1, note that from (22) we obtain

T_{ε}(ϕ) ∫_{−1}^{1} Ψ(T_{ε}(ϕ)ε, ϕ, s) dμ(s) + ∫_{−1}^{1} Φ(T_{ε}(ϕ)ε, ϕ, s) dμ(s) = 0.

Differentiating this equation with respect to ϕ and denoting by Φ_{i} and Ψ_{i} the derivatives of Φ and Ψ with respect to the ith variable, i = 1, 2, we get

T′_{ε}(ϕ) ∫_{−1}^{1} Ψ dμ(s) + T_{ε}(ϕ) ∫_{−1}^{1}(Ψ_{1}T′_{ε}(ϕ)ε + Ψ_{2}) dμ(s) + ∫_{−1}^{1}(Φ_{1}T′_{ε}(ϕ)ε + Φ_{2}) dμ(s) = 0.

This yields

(25)  T′_{ε}(ϕ) = −\frac{∫_{−1}^{1} (T_{ε}(ϕ)Ψ_{2} + Φ_{2}) dμ(s)}{∫_{−1}^{1} (Ψ + ε(T_{ε}(ϕ)Ψ_{1} + Φ_{1})) dμ(s)}.

We have already shown that T_{ε}(ϕ), ρ_{ε}, and the partial derivatives of ρ_{ε} with respect to ϕ of any order are each bounded uniformly in ε for 0 < ε < ε_{1}(n). Furthermore we have that ρ_{ε} is bounded below by a positive constant, uniformly in ε. It then follows from (23) and (24) that the partial derivatives Φ_{i} and Ψ_{i}, i = 1, 2, are all bounded uniformly in ε. From (24) we also deduce that Ψ is bounded below by a positive constant, uniformly in ε. These facts and (25) show that T′_{ε}(ϕ) is bounded uniformly in ε, proving (12) for k = 1.

Differentiation of (23) and (24) with respect to the variables t_{ε}(ϕ) and ϕ, together with the previously established facts just mentioned, reveals that the partial derivatives of Φ(t_{ε}(ϕ), ϕ, s) and Ψ(t_{ε}(ϕ), ϕ, s) with respect to its first and second variables, of any order, are each bounded uniformly in ε for 0 < ε < ε_{1}(n). This and the fact that Ψ is bounded below by a positive constant, uniformly in ε, allow the proof of (iii) to be completed by repeatedly differentiating (25) with respect to ϕ and using induction on k. □

Lemma 3.4. Let n ≥ 3, let K_{ε} be as in Lemma 3.2, and let 0 < ε < ε_{1}(n). Then m_{K_{ε}} ∈ C_{e}^{∞}(S^{n−1}) and there is a rotationally symmetric function g_{ε} ∈ C_{e}^{∞}(S^{n−1}) such that

(26)  (n − 1)(R^{−1}m_{K_{ε}})(u) = 1 + (R^{−1}g_{ε})(u)ε,

for u ∈ S^{n−1}, where R is the spherical Radon transform. Moreover, there is a constant c_{2}(k, n) such that

(27)  |\frac{d^{k}g_{ε}(ϕ)}{dϕ^{k}} | ≤ c_{2}(k, n),

for all 0 < ε < ε_{1}(n) and ϕ ∈ S^{1}.

Proof. Let 0 < ε < ε_{1}(n). In the notation of the previous lemma, using (20), we have

(28)  m_{K_{ε}}(ϕ) = max_{t∈ℝ}A_{K_{ε},ϕ}(t) = A_{K_{ε},ϕ}(t_{ε}(ϕ)) = \frac{ω_{n−2}}{n − 1} ∫_{−1}^{1} ρ_{ε}(t_{ε}(ϕ), ϕ, s)^{n−1} dμ(s),

for ϕ ∈ S^{1}. We know from the paragraph after (15) that ρ_{ε}(t, ϕ, s) is infinitely differentiable with respect to each of its variables and also that t_{ε}(ϕ) ∈ C^{∞}(S^{1}) by Lemma 3.3(ii). It follows that m_{K_{ε}}(ϕ) ∈ C_{e}^{∞}(S^{1}) and therefore m_{K_{ε}}(u) ∈ C_{e}^{∞}(S^{n−1}).

We wish to rewrite the previous equation in a more convenient form. To this end, we first use (15) to obtain

1 − ρ_{ε} = \frac{(t cos ϕ + ρ_{ε}s sin ϕ)^{3}(1 + (t^{2} + ρ_{ε}^{2})^{3/2})}{(1 + ρ_{ε})(1 + t^{2} + ρ_{ε}^{2} + (t^{2} + ρ_{ε}^{2})^{2})}ε + \frac{t^{2}}{1 + ρ_{ε}} .

When t = t_{ε}(ϕ), this and (11) imply that

ρ_{ε}(t_{ε}(ϕ), ϕ, s) = 1 − U_{ε}(ϕ, s)ε,

say, where

(29)  U_{ε}(ϕ, s) = \frac{(T_{ε}(ϕ)ε cos ϕ + ρ_{ε}s sin ϕ)^{3} (1 + (T_{ε}(ϕ)^{2}ε^{2} + ρ_{ε}^{2})^{3/2})}{(1 + ρ_{ε}) (1 + T_{ε}(ϕ)^{2}ε^{2} + ρ_{ε}^{2} + (T_{ε}(ϕ)^{2}ε^{2} + ρ_{ε}^{2})^{2})} + \frac{T_{ε}(ϕ)^{2}ε}{1 + ρ_{ε}}.

Therefore

(30)  ρ_{ε}(t_{ε}(ϕ), ϕ, s)^{n−1} = 1 + f_{ε}(ϕ, s)ε,

say, where

(31)  f_{ε}(ϕ, s) = ∑_{j=1}^{n−1}(−1)^{j}\binom{n − 1}{j}U_{ε}(ϕ, s)^{j}ε^{j−1}.

By Lemma 3.3(iii), T_{ε}(ϕ) and all its derivatives are bounded above uniformly in ε. Also, we showed in the proof of Lemma 3.3(ii) (paragraph after (15)) that ρ_{ε} and all its derivatives with respect to ϕ are bounded uniformly in ε, and that ρ_{ε} is bounded below by a positive constant, uniformly in ε. These facts and repeated differentiation with respect to ϕ of (31) and (29) show that for every k = 0, 1, …, there is a constant a(k, n) such that

(32)  |\frac{d^{k}f_{ε}(ϕ, s)}{dϕ^{k}} | ≤ a(k, n),

for all 0 < ε < ε_{1}(n), all ϕ ∈ S^{1}, and all −1 ≤ s ≤ 1.

From (28) and (30), we obtain

(33)  A_{K_{ε}, ϕ}(t_{ε}(ϕ)) = \frac{ω_{n−2}}{n − 1} ∫_{−1}^{1} (1 + f_{ε}(ϕ, s)ε) dμ(s).

Note that when ε = 0, K_{ε} = B^{n}, t_{ε}(ϕ) = 0, and A_{K_{ε}, ϕ}(t_{ε}(ϕ)) = κ_{n−1}. Hence, or by direct integration, we obtain

(34)  \frac{ω_{n−2}}{n − 1} ∫_{−1}^{1} 1 dμ(s) = κ_{n−1}.

Let

(35)  g_{ε}(u) = g_{ε}(ϕ) = ω_{n−2} ∫_{−1}^{1} f_{ε}(ϕ, s) dμ(s),

for u ∈ S^{n−1} and ϕ ∈ S^{1}. From (33) and (34) we obtain

m_{K_{ε}}(u) = max_{t∈ℝ}A_{K_{ε},u}(t) = κ_{n−1} + g_{ε}(u)ε/(n − 1),

for all u ∈ S^{n−1}. Now g_{ε}(u) ∈ C_{e}^{∞}(S^{n−1}) follows from m_{K_{ε}}(u) ∈ C_{e}^{∞}(S^{n−1}). We also have

(n − 1)(R^{−1}m_{K_{ε}})(u) = R^{−1} ((n − 1)κ_{n−1}) + (R^{−1}g_{ε})(u)ε = 1 + (R^{−1}g_{ε})(u)ε,

for all u ∈ S^{n−1}. Finally, (27) follows directly from (32) and (35). □

Lemma 3.5. Let p ∈ {1, …, n − 1} and let g ∈ C^{∞}(ℝ^{n} \ {o}) be homogeneous of degree −n + p. Let

G_{u}(z) = (1 − z^{2})^{(n−3)/2} ∫_{S^{n−1}∩u^{⊥}} g(zu + \sqrt{1 − z^{2}} v) dv,

for −1 ≤ z ≤ 1. If g is even, then

ĝ(u) = \{ \begin{matrix}(−1)^{(p−1)/2}πG_{u}^{(p−1)}(0), & if p is odd, \\ (−1)^{p/2}(p − 1)! (∫_{−1}^{1} |z|^{−p}(G_{u}(z) − ∑_{k=0}^{p−1} G_{u}^{(k)}(0)z^{k}/k!) dz+ \\ +2 ∑\{G_{u}^{(k)}(0)/(k!(1 + k − p)) : k = 0, …, p − 2, k even\}), & if p is even, \end{matrix}

for u ∈ S^{n−1}. If g is odd, then

ĝ(u) = \{ \begin{matrix}(−1)^{p/2}πG_{u}^{(p−1)}(0)i, & if p is even, \\ (−1)^{(p+1)/2}(p − 1)! (∫_{−1}^{1} |z|^{−p} sgn z(G_{u}(z) − ∑_{k=0}^{p−1} G_{u}^{(k)}(0)z^{k}/k!) dz+ \\ +2 ∑\{G_{u}^{(k)}(0)/(k!(1 + k − p)) : k = 1, …, p − 2, k odd\})i, & if p is odd, \end{matrix}

for u ∈ S^{n−1}.

Proof. The Fourier transform of a real even function is real and the Fourier transform of a real odd function is purely imaginary. Using these well-known facts, the four formulas for ĝ(u) in the statement of the lemma are obtained by taking real or imaginary parts, as appropriate, in the more general formulas [8, Equations (3.6) and (3.7), p. 395], which apply to any function g ∈ C^{∞}(S^{n−1}), extended to a homogeneous of degree −n + p function, p ∈ {1, …, n − 1}, on ℝ^{n} \ {o} as in (2) with f = g. □

Lemma 3.6. Let n ≥ 3 and let K_{ε} be as in Lemma 3.2. For 0 < ε < ε_{1}(n), define

(36)  ρ_{L_{ε}}(u) = ((n − 1)(R^{−1}m_{K_{ε}})(u))^{1/(n−1)} ,

for all u ∈ S^{n−1}. Then there is a positive ε_{2}(n) ≤ ε_{1}(n) such that ρ_{L_{ε}} is the radial function of an origin-symmetric star body L_{ε} in ℝ^{n} with ρ_{L_{ε}} ∈ C^{∞}(S^{n−1}), for all 0 < ε < ε_{2}(n).

Proof. Let 0 < ε < ε_{1}(n) and let g_{ε} ∈ C_{e}^{∞}(S^{n−1}) be the function from Lemma 3.4. Extend g_{ε} to a homogeneous of degree −1 function in C_{e}^{∞}(ℝ^{n} \ {o}) as in (2) with f = g_{ε} and p = n − 1. Then, by (7), we have

(R^{−1}g_{ε})(u) = \frac{π}{(2π)^{n}} ĝ_{ε}(u),

for u ∈ S^{n−1}. From this, (26), and (36), we conclude that

(37)  ρ_{L_{ε}}(u) = (1 + \frac{π}{(2π)^{n}} ĝ_{ε}(u)ε)^{1/(n−1)} ,

for all u ∈ S^{n−1}.

Define

(38)  G_{ε,u}(z) = (1 − z^{2})^{(n−3)/2} ∫_{S^{n−1}∩u^{⊥}} g_{ε}(zu + \sqrt{1 − z^{2}} v) dv,

for −1 ≤ z ≤ 1. Since (27) holds for all ϕ ∈ S^{1} (and not just for ϕ ∈ [0, π]), it is not difficult to check that the rotationally symmetric function g_{ε}(x/|x|), x ∈ ℝ^{n} \ {o}, has each of its partial derivatives with respect to x_{i}, i = 1, …, n, of any order bounded on S^{n−1}, uniformly in ε. From this and (38) we see that for k = 0, 1, …, there are constants c_{3}(k, n) such that

(39)  |G_{ε,u}^{(k)}(0)| ≤ c_{3}(k, n),

for all 0 < ε < ε_{1}(n) and u ∈ S^{n−1}.

Next, we apply Lemma 3.5 for the case when g is even (with g and G_{u} replaced by g_{ε} and G_{ε,u}, respectively). Suppose first that n is even. By (39) and Lemma 3.5 with p = n − 1, it follows that ĝ_{ε}(u) = (−1)^{(n−2)/2}πG_{ε,u}^{(n−2)}(0) is bounded on S^{n−1}. From (37) we conclude that there is a positive ε_{2}(n) ≤ ε_{1}(n) such that ρ_{L_{ε}}(u) > 0 for all u ∈ S^{n−1}, and hence such that L_{ε} is an origin-symmetric star body, for all 0 < ε < ε_{2}(n).

Now suppose that n is odd. By Lemma 3.5 with p = n−1, we obtain ĝ_{ε}(u) = (−1)^{(n−1)/2}(n− 2)!(I_{ε}(u) + Σ_{ε}(u)), where

I_{ε}(u) = ∫_{−1}^{1} |z|^{−n+1} (G_{ε,u}(z) − ∑_{k=0}^{n−2} G_{ε,u}^{(k)}(0)\frac{z^{k}}{k!} ) dz

and

Σ_{ε}(u) = 2 ∑ \{\frac{G_{ε,u}^{(k)}(0)}{k!(2 + k − n)} : k = 0, …, n − 3, k even\} .

By (39), Σ_{ε}(u) is bounded on S^{n−1}. The integral I_{ε}(u) over the range [0, 1] can be written as (40)

I_{ε}^{+}(u) = ∫_{0}^{1/2} z^{−n+1} (G_{ε,u}(z) − ∑_{k=0}^{n−2} G_{ε,u}^{(k)}(0)\frac{z^{k}}{k!} ) dz+∫_{1/2}^{1} z^{−n+1} (G_{ε,u}(z) − ∑_{k=0}^{n−2} G_{ε,u}^{(k)}(0)\frac{z^{k}}{k!} ) dz.

Using (39) again, we see that the second integral in (40), as a function of u, is bounded on S^{n−1}. By Taylor’s theorem, the first integral in (40) is equal to

∫_{0}^{1/2} \frac{G_{ε,u}^{(n−1)}(c(z))}{(n − 1)!} dz,

for some 0 < c(z) < 1/2. It follows from (38) that as a function of u, G_{ε,u}^{(n−1)}(z) is bounded on S^{n−1}, uniformly in z for 0 ≤ z ≤ 1/2. Therefore I_{ε}^{+}(u) is bounded on S^{n−1}. Similarly, the integral I_{ε}(u) over the range [−1, 0] is bounded on S^{n−1}. Consequently, I_{ε}(u) itself and therefore ĝ_{ε}(u) are bounded on S^{n−1}. Then, as for the case when n is even, we conclude from (37) that there is a positive ε_{2}(n) ≤ ε_{1}(n) such that L_{ε} is an origin-symmetric star body, for all 0 < ε < ε_{2}(n). □

Lemma 3.7. Let L_{ε} be as in Lemma 3.6, where 0 < ε < ε_{2}(n). Then there is a positive ε_{3}(n) ≤ ε_{2}(n) such that L_{ε} is convex for all 0 < ε < ε_{3}(n).

Proof. Let 0 < ε < ε_{2}(n) and let g_{ε} be as in Lemma 3.6. Since L_{ε} is a body of revolution whose radial function is given by (37), its intersection with the x_{1}, x_{n}-plane is a curve whose radial function is

(41)  r(ϕ) = (1 + \frac{π}{(2π)^{n}} ĝ_{ε}(sin ϕ e_{1} + cos ϕ e_{n}) ε)^{1/(n−1)} .

It suffices to prove that this curve is convex for small ε.

We claim that the first and second partial derivatives of ĝ_{ε} with respect to x_{1} and x_{n} are bounded on S^{n−1} uniformly in ε for small ε. Once this has been proved, it follows that the first and second derivatives of ĝ_{ε}(sin ϕ e_{1} + cos ϕ e_{n}) with respect to ϕ are bounded uniformly in ε for small ε. The required convexity is then an easy consequence of the polar coordinate formula (10) for curvature, (41), and the fact that r(ϕ) is close to 1 when ε is small.

It remains to prove the claim above. By [12, Lemma 3.16], ĝ_{ε} is an infinitely differentiable function that is homogeneous of degree −n+1 on ℝ^{n}\{o}. Therefore for q = 1, …, n, ∂ĝ_{ε}/∂x_{q} is an infinitely differentiable function that is homogeneous of degree −n on ℝ^{n} \ {o}.

Let ψ be a test function with support in ℝ^{n} \ {o} and let q ∈ {1, …, n}. Denote by ⟨∂ĝ_{ε}/∂x_{q}, ψ⟩ the action of the distribution ∂ĝ_{ε}/∂x_{q} and note that since ψ has support in ℝ^{n} \ {o}, the derivative ∂ĝ_{ε}/∂x_{q} can be regarded in both the distributional and the usual sense. Let h_{ε,q}(x) = x_{q}g_{ε}(x) for x ∈ ℝ^{n} \ {o}. Using first (4) with f = g_{ε} and then (5) with f = h_{ε,q}, we obtain

⟨\frac{∂ĝ_{ε}}{∂x_{q}} , ψ⟩ = −i⟨ĥ_{ε,q}, ψ⟩ = −i⟨|x|^{2}ĥ_{ε,q}(x), |x|^{−2}ψ(x)⟩
= i⟨{ \atop Δh_{ε,q}}(x), |x|^{−2}ψ(x)⟩ = i⟨|x|^{−2}{ \atop △h_{ε,q}}(x), ψ(x)⟩.

This gives

(42)  \frac{∂ĝ_{ε}}{∂x_{q}} = i|x|^{−2}{ \atop △h_{ε,q}}

on ℝ^{n} \{o}, because by [15, Theorem 6.25], distributions that are equal on test functions with support in ℝ^{n} \ {o} can differ at most by derivatives of the delta function.

Let g(x) = △h_{ε,q}(x) = △(x_{q}g_{ε})(x) for x ∈ ℝ^{n} \ {o}. Then, with (42) in hand, the desired bounds for ∂ĝ_{ε}/∂x_{q} on S^{n−1} will follow from corresponding bounds for ĝ. To this end, observe that since x_{q}g_{ε} is an infinitely differentiable odd function on ℝ^{n} \ {o} that is homogeneous of degree 0, g is an infinitely differentiable odd function on ℝ^{n} \ {o} that is homogeneous of degree −2. Therefore ĝ is an infinitely differentiable function on ℝ^{n} \{o} that is homogeneous of degree −n + 2 and which can be computed by Lemma 3.5. Letting

G_{u}(z) = (1 − z^{2})^{(n−3)/2} ∫_{S^{n−1}∩u^{⊥}} g(zu + \sqrt{1 − z^{2}} v) dv,

for −1 ≤ z ≤ 1, and using (27), we see that for k = 0, 1, …, there are constants c_{4}(k, n) such that

(43)  |G_{u}^{(k)}(0)| ≤ c_{4}(k, n),

for all 0 < ε < ε_{2}(n) and u ∈ S^{n−1}. Bounds for ĝ now follow from Lemma 3.5 for the case when g is odd and p = n − 2. Indeed, it follows immediately from this and (43) that when n is even, ĝ(u) = (−1)^{(n−2)/2}πG_{u}^{(n−3)}(0) is bounded on S^{n−1} uniformly in ε for small ε. The case when n is odd is handled in exactly the same way as in Lemma 3.6 and we omit the details.

Bounds for the second partial derivatives of ĝ_{ε} on S^{n−1} are obtained in an entirely analogous fashion, as follows. Let q, r ∈ {1, …, n} and let j_{ε,q,r}(x) = x_{q}x_{r}g_{ε}(x) for x ∈ ℝ^{n} \ {o}. An argument similar to the one that showed (42) gives

\frac{∂^{2}ĝ_{ε}}{∂x_{q}∂x_{r}} = |x|^{−2}{ \atop △j_{ε,q,r}}

on ℝ^{n} \ \{o\}. Here { \atop △j_{ε,q,r}} is an infinitely differentiable even function on ℝ^{n} \ \{o\} that is homogeneous of degree −1. Bounds for { \atop △j_{ε,q,r}} on S^{n−1} are then obtained via (43) and Lemma 3.5 for the case when g is even and p = n − 1. Again, we omit the details since the argument is exactly the same as for the first partial derivatives of ĝ_{ε}. □

Proof of Theorem 3.1. Let n ≥ 3 and 0 < ε < ε_{3}(n). Let K = K_{ε} be defined by (8) and let L = L_{ε} be defined by (36). By Lemmas 3.2 and 3.7, K and L are convex bodies of revolution about the x_{n}-axis. Lemma 3.2 shows that ρ_{K} ∈ C^{∞}(S^{n−1}) and that K is not centrally symmetric. From Lemma 3.6, we get that ρ_{L} ∈ C^{∞}(S^{n−1}) and that L is origin symmetric. The Brunn-Minkowski theorem implies that m_{L}(u) = V (L ∩ u^{⊥}) for all u ∈ S^{n−1}. By Lemma 3.4, we have m_{K} ∈ C_{e}^{∞}(S^{n−1}). Therefore, by (36) and the fact that R : C_{e}^{∞}(S^{n−1}) → C_{e}^{∞}(S^{n−1}) is a continuous bijection, we obtain

m_{L}(u) = V (L ∩ u^{⊥}) = \frac{1}{n − 1}(Rρ_{L}^{n−1})(u) = (R(R^{−1}m_{K}))(u) = m_{K}(u),

for all u ∈ S^{n−1}. □

4. Concluding remarks

The definition (8) of the body K = K_{ε} appears to be essentially the simplest that allows our construction to work. For example, if we define

ρ_{K_{ε}}(ϕ) = (1 + ε cos ϕ)^{−1} ,

where 0 < ε < 1 and 0 ≤ ϕ ≤ π is the angle with the positive x_{n}-axis, then K = K_{ε} is an ellipsoid. In particular, it is centrally symmetric. If K′ denotes the origin-symmetric translation of K, then our construction would yield IL = CK = CK′ = IK′. This would imply K′ = L and hence that K and L are equal up to a translation. Of course the formula

ρ_{K_{ε}}(ϕ) = (1 + ε cos^{2} ϕ)^{−2} ,

gives a K = K_{ε} that is already origin symmetric.

Klee [10], [11] (see also [2, p. 24] and [6, Problem 8.8(ii)]) also asked whether a convex body in ℝ^{n}, n ≥ 3, whose inner section function is constant, must be a ball. The hypothesis is actually weaker than that of Bonnesen’s question in [1] of whether a convex body in ℝ^{n}, n ≥ 3, must be a ball if both its inner section function and its brightness function are constant. Today, Bonnesen’s question (see [2, p. 24] and [6, Problem 8.9(ii)]) is one of the oldest open problems in convex geometry. (For n = 2, a counterexample is again provided by any convex body of constant width that is not a disk; this follows from a result of E. Makai and H. Martini (see [6, Theorem 8.3.5]) that in the plane, the cross-section body and projection body coincide.)

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Department of Mathematics, Western Washington University, Bellingham, WA 98225-9063

E-mail address: Richard.Gardner@wwu.edu

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

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

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

E-mail address: vladyaskin@math.ualberta.ca

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

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