Symmetry from sectional integrals for convex domains

Abstract

Let Ω be a bounded convex domain in ${\mathbb{R}}^n$ ($n\ge 2$). In this work, we prove that if there exists an integrable function f such that it's Radon transform over $(n-1)$-dimensional hyperplanes intersecting the domain Ω is a function G depending on the distance to the nearest parallel supporting hyperplane to Ω, then Ω is a ball and f is radial depending on certain assumptions on G. As a consequence we show that constants are not in range of Radon transform of integrable functions in dimension $n\ge 3$.

Introduction

Let Ω be a domain in ${\mathbb{R}}^n$ and f be an integrable function in Ω. We extend f to be identically 0 outside of Ω. Let $〈,〉$ denote the usual inner product in ${\mathbb{R}}^n$. Throughout the article ${\mathcal{H}}^k⌞G$ denotes the k-dimensional Hausdorff measure restricted to G, a Borel measurable subset of ${\mathbb{R}}^n$ for $1\le k\le n$.

The ray transform integrates scalar functions over straight lines. The family of oriented lines can be parametrized by the points on the manifold

$$T{\mathbb{S}}^{n-1}=\{(x,\xi )\in {\mathbb{R}}^n\times {\mathbb{R}}^n:〈x,\xi 〉=0,|\xi |=1\}\subset {\mathbb{R}}^n\times {\mathbb{R}}^n$$

which is the tangent bundle of the unit sphere. The ray transform I of an integrable function f is a function defined on $T{\mathbb{S}}^{n-1}$ as

$$If(x,\xi )=\underset{-\infty }{\overset{+\infty }{\int }}f(x+t\xi )dt.$$

The Radon transform integrates the functions over the hyperplanes. The Radon transform R of a function $f\in L^1(\mathrm{\Omega })$ is the function defined on ${\mathbb{S}}^{n-1}\times \mathbb{R}$ by

$$Rf(w,p)=\underset{{\mathrm{\Sigma }}_{\omega ,p}}{\int }f(x)d{\mathcal{H}}^{n-1}⌞{\mathrm{\Sigma }}_{\omega ,p}$$

where ${\mathrm{\Sigma }}_{w,p}=\{x\in {\mathbb{R}}^n|〈x,\omega 〉=p\}$ denotes the hyperplane with p as the perpendicular distance from the origin and ω is normal to the plane. $If{|}_{\mathrm{\Omega }}$ denotes the ray transform of f along all the lines intersecting the domain Ω. Similar definition stands for the notation $Rf{|}_{\mathrm{\Omega }}$. The operators (1.1) and (1.2) have been well studied and has many applications in computer tomography. For more detailed study of the operators I and R we refer [3], [5].

In dimension $n=2$ both operators I and R coincide. The characterization of range of these operators have been well studied in case of Schwartz class functions $\mathcal{S}({\mathbb{R}}^n)$. It is known that I and R are linear isomorphisms between the Schwartz spaces $\mathcal{S}({\mathbb{R}}^n)$ to $\mathcal{S}(T{\mathbb{S}}^{n-1})$ and $\mathcal{S}({\mathbb{S}}^{n-1}\times \mathbb{R})$ respectively. Both the operators I and R are injective, i.e., if $If=0$ imply $f=0$ and $Rf=0$ imply $f=0$.

The question one is interested in is,

Question 1

Are non-zero constant functions in the range of the ray transforms, i.e., does $\exists f\in L^1(\mathrm{\Omega })$ such that $If{|}_{\mathrm{\Omega }}=c(\ne 0)$?

In [4] the authors showed that in Euclidean spaces of dimension $n\ge 2$, smooth strictly convex domains Ω admitting functions of constant X-ray transform are balls and the function is unique and radial. The function $f(x)=\frac{{\chi }_{\{|x|<R\}}}{\pi \sqrt{R^2-|x{|}^2}}$ supported in the ball of radius R centered at the origin of ${\mathbb{R}}^n$ has constant X-ray transform in its support in ${\mathbb{R}}^n$.

Similarly, one can ask the same question for the Radon transform in dimensions $n\ge 2$.

Question 2

Are non-zero constant functions in the range of Radon transforms in dimension $n\ge 2$, i.e., does $\exists f\in L^1(\mathrm{\Omega })$ such that $Rf{|}_{\mathrm{\Omega }}=c(\ne 0)$?

We address this question in section 2 and show that one cannot expect constants to be in the range of Radon transform in dimensions $n\ge 3$ by proving the following analogue. If Ω is a bounded convex domain in ${\mathbb{R}}^n$ ($n\ge 2$) and if $\exists f\in L^1(\mathrm{\Omega })$ such that $Rf{|}_{\mathrm{\Omega }}$ is a function G (see Theorem 2.1 for precise definition) that depends only on the distance to nearest parallel supporting hyperplane to Ω and does not vanish identically on any interval [see Theorem 2.1 for precise definition], then Ω is a ball and f is a unique radial function about the center of Ω.

The proofs utilize the integral moments of the function similar to the case of the ray transform in [4].

Subsequently, as a corollary to the main result in our paper, we obtain the result for the ray transform $If{|}_{\mathrm{\Omega }}=c(\ne 0)$ in Euclidean space under milder assumptions requiring Ω to be a bounded domain in ${\mathbb{R}}^2$ or a bounded convex domain in ${\mathbb{R}}^n$ for $n\ge 3$. To our knowledge, the main result in Section 2 is completely new. Recently a related result has been proved in [1] that if the Radon transform of a distribution is supported on tangent planes to ∂Ω for a bounded convex domain Ω, then ∂Ω is an ellipsoid.

Remark 1.1

If Ω is bounded convex domain in ${\mathbb{R}}^n$, any line ℓ intersecting Ω intersects ∂Ω at exactly two points. Also if $p\in \partial \mathrm{\Omega }$ then there exists at least one supporting $(n-1)$-dimensional hyperplane of Ω passing through p.