Local fixed point results for centroid body operators

Characterizing the Euclidean space among all real normed spaces is one of the aims of the ten problems formulated in 1956 by Busemann and Petty [4]. These problems lead to the study of certain operators on convex bodies in ${\mathbb{R}^{n}}$, such as the intersection body operator for the first Busemann-Petty problem. Proving a particular property of the intersection body operator in each dimension resulted in the resolution of the first Busemann-Petty problem in 1996. In the class of convex bodies, proving properties for operators globally is challenging. The local study of such operators on convex bodies appears to be a more approachable initial step. In [5], it was established that, other than linear transformations of the Euclidean ball, there are no additional fixed points of the intersection body operator in a neighborhood of the Euclidean ball. A similar result was obtained in [12] for the projection body operator. In [1], a local affirmative answer was obtained for the Busemann-Petty problems 5 and 8, by studying the local fixed points of the polar-intersection body operator.

In this paper we conduct a similar local study for several problems involving the centroid body operator. First, we consider its fixed points.
Problem 1. If for an origin-symmetric convex body $K\subset{\mathbb{R}^{n}}$, $n\geq 3$ we have

where $\Gamma K$ is the centroid body of $K$ and $c$ is some constant, need $K$ be an ellipsoid?

Below we will verify that the ball does indeed satisfy this condition, and that equation (1) is invariant under linear transformations. Consequently, this equation holds for all ellipsoids. We prove the following result.

A similar result is obtained for the $p$-centroid body operators $\Gamma_{p}$, $p\in(1,+\infty)$.

In addition, we prove a local result involving the $p$-centroid body and the polar intersection body of $K$:

Theorem 3 is similar to several open problems listed by Gardner in [6, pg 336-337], relating the intersection body of $K$ with the difference body $\Delta K$ and the cross section body $CK$ of $K$:

Problem 8.1 Suppose that $K$ is a convex body such that $\Delta K$ and ${\mathcal{I}}^{*}K$ are homothetic. Must $K$ be a centered ellipsoid?

Problem 8.9 (ii) If $\Pi K$ is homothetic to either ${\mathcal{I}}K$ or $CK$, is $K$ an ellipsoid?

In two dimensions, the answer to Problem 1 is known to be affirmative: the only convex bodies which are dilates of their centroid bodies are ellipses. This was proven by Petty in [9, Theorem 5.6]. Indeed, the centroid body of $K$ has an important role in fluid statics. If $K$ is an origin-symmetric convex body with density 1/2 (with water having density 1), each point on the boundary of the centroid body of $K$ is the center of mass of the underwater part of $K$ in a given direction, and $\Gamma K$ is the convex hull of the locus of all such centers of mass in all directions. The flotation properties of $K$ are governed by three Theorems of Dupin for $\Gamma K$ (see for example [11]). In particular, in two dimensions, the Third Theorem of Dupin states that the radius of curvature $R(\xi)$ of the centroid body $\Gamma K$ at a boundary point perpendicular to a unit direction $\xi$ is proportional to the cube of the length of the chord $\ell(\xi)=K\cap\xi^{\perp}$,

If we assume that $\Gamma K$ is homothetic to $K$, then the same relation is true between the curvature of $K$ at a point and the cube of the length of the corresponding central section. But in the two dimensional case this is a restatement of the Busemann-Petty Problem number 8 in dimension 2 [4, 9].

Let ${\mathcal{S}^{n-1}}=\{x\in\mathbb{R}^{n}:|x|=1\}$ be the unit sphere in $\mathbb{R}^{n}$, where $|x|$ denotes the Euclidean norm. Let $\sigma_{n-1}$ be the spherical Lebesgue measure on ${\mathcal{S}^{n-1}}$ and $\lambda_{n}$ be the $n$-dimensional Lebesgue measure. A convex body $K$ is a convex compact subset of $\mathbb{R}^{n}$ with non-empty interior. We will assume that the origin is an interior point of convex bodies. $K$ is origin symmetric if for every $x\in K$, $-x\in K$.

Given a convex body $K\subset{\mathbb{R}^{n}}$, the support function $h_{K}$ of $K$ is defined as

and its radial function is

A body $L$ is called the centroid body of a convex body $K$ if

The body $L$ is denoted as $\Gamma K$. Switching this integral to polar coordinates and simplifying yields

Examining the right hand side, we observe that it is a multiple of the cosine transform. The cosine transform ${\mathcal{C}}f$ of an integrable function $f:{\mathcal{S}^{n-1}}\rightarrow{\mathbb{R}}$ is defined as

Using this to rewrite the support function of the centroid body gives

Equation (1) can now be rewritten analytically as

A body $L$ is called the intersection body of the convex body $K$ if

The body $L$ is denoted as ${\mathcal{I}}K$. Expressing the volume as an integral and passing to polar coordinates, we obtain

The right hand side is almost the spherical Radon transform. The spherical Radon transform of an integrable function $f\in L^{1}({\mathcal{S}^{n-1}})$ is denoted ${\mathcal{R}}f$ and is defined as

Thus, the radial function of the intersection body can be expressed in terms of the Radon transform of the radial function of the original body, by

A body $L$ is the polar body of a convex body $K$ if

The body $L$ is denoted by $K^{*}$. The polar intersection body of a body $K$ is ${\mathcal{I}}^{*}K=({\mathcal{I}}K)^{*}$. Hence,

Combining this with equation (4) gives the reformulation of $c\Gamma K={\mathcal{I}}^{*}K$ as the following equation,

To simplify the exposition, we will first consider the case $p=1$. The analytic reformulation of equation (2) will be derived in Section 5, and the analytic reformulation of (3) will be considered in Section 6.

Remark: Even though Theorem 1 does not make such assumptions on the body $K$, if $K$ satisfies (1), then it must be centrally symmetric, strictly convex and smooth, since $\Gamma K$ has those properties (see [6, Sec. 9.1]).

Spherical harmonics are homogeneous harmonic polynomials restricted to the sphere. They form the eigenspaces for a collection of common linear operators, such as the spherical Radon transform, the cosine transform and the Laplace-Baltrami operator on the sphere. In this section, we will mostly follow the expositions from Groemer [7] and Atkison-Han [2].

The space of all harmonic, homogeneous polynomials of degree $k$ on $n$ variables whose domain is restricted to the sphere is denoted ${{\mathcal{H}}_{k}^{n}}$. Its elements are called the spherical harmonics of degree $k$. The spherical harmonic spaces are orthogonal with respect to the $L^{2}({\mathcal{S}^{n-1}})$ inner product. The space of all finite sums of spherical harmonics on $n$ variables is denoted by ${{\mathcal{H}}^{n}}=\oplus_{k=0}^{\infty}{{\mathcal{H}}_{k}^{n}}$, and called the space of spherical harmonics.

The space of spherical harmonics is dense in $L^{2}({\mathcal{S}^{n-1}})$ and by extension also in $C({\mathcal{S}^{n-1}})$. The orthogonal projection onto the spherical harmonics of degree $k$, will be denoted ${\mathcal{P}_{k}}:C({\mathcal{S}^{n-1}})\rightarrow{{\mathcal{H}}_{k}^{n}}$ (see [2, Def. 2.11] for an explicit definition of the projection). The eigenvalues of the cosine transform on each of the spherical harmonic spaces are (see [7, Lemmas 3.4.1, 3.4.5)]), for $k=0$, ${\mathcal{C}}{\mathcal{P}_{0}}=2\kappa_{n-1}{\mathcal{P}_{0}}$ and for even $k>0$

where $(k+2)!!=(k+2)(k!!)$. Since the cosine transform is defined as an integral over a symmetric domain, ${\mathcal{C}}{\mathcal{P}_{k}}=0$ when $k$ is odd.

When the body $K$ is placed in the isotropic position, we have the following estimate of the second harmonic of $\rho_{K}$, obtained in [1, Sec. 9]. Let $r_{0}={\mathcal{P}_{0}}\rho_{K}=\int_{{\mathcal{S}^{n-1}}}\rho_{K}d\sigma$ be the mean value of $\rho_{K}$ on the sphere, and let $\rho_{K}-r_{\scriptscriptstyle{0}}={\mathcal{P}_{2}}(\rho_{K})+{\mathcal{P}_{4}}(\rho_{K})+\dots$ be the spherical harmonic decomposition of $\rho_{K}-r_{\scriptscriptstyle{0}}$. From the definition (10) of the isotropic position, we have that if $p(x)$ is a quadratic harmonic polynomial, $p(x)=\sum\limits_{i,j}a_{ij}x_{i}x_{j}$ with $\sum\limits_{i=1}^{n}a_{ii}=0$, then

This means $\rho_{K}^{n+2}$ has no second order term in its spherical harmonic decomposition. If $K$, additionally, was also close to the Euclidean ball in the sense $\left\lvert\rho_{K}-r_{0}\right\rvert<\epsilon$, then ${\mathcal{P}_{2}}\rho_{K}$ should also be small in the $L^{2}({\mathcal{S}^{n-1}})$ sense. Indeed, estimating the error of the first order Taylor polynomial of $(x+r_{0})^{n+2}$ at $\rho_{K}-r_{0}$, yields

The second harmonic ${\mathcal{P}_{2}}(\rho_{K})$ of $\rho_{K}$ can now be estimated from $\rho_{K}^{n+2}$ using the previous equation. Applying the $L^{2}({\mathcal{S}^{n-1}})$ to each side and using the orthogonality of the spherical harmonic spaces, we obtain

which yields that

This section introduces $p$-centroid body and extends the result of the previous section to $p>1$. Subsection 5.1 gives the definition of the $p$-centroid body and an analytic reformulation in terms of the $p$-cosine transform. As the differences between the prior case and the $p>1$ are purely technical, the subsequent subsections include relevant details for the extension of Theorem 1 to $p>1$ for completeness.