Questions tagged [integral-geometry]

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2
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45 views

Radon transform range theorem and radial functions

(UPDATED for rapid decay considerations + new question) In dimension 2, the Radon transform range theorem states that a rapidly decaying (Schwartz) function $g(t,\theta)$ can be represented as a ...
2
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0answers
70 views

The mean along the eccentric anomaly of an ellipse log distance to a point within the ellipse

Conjecture. Let $$ f(r,\alpha,p, \theta) = \ln\left(\left(r\sin\alpha-\sin\theta\right)^{2}\left(1-p\right)^{2}+\left(r\cos\alpha-\cos\theta\right)^{2}\left(1+p\right)^{2}\right). $$ Then for any ...
3
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0answers
76 views

Radon transform on complex Grassmannians

Let $Gr_{i,n}$ denote the Grassmannian of complex linear $i$-dimensional subspaces in the Hermitian space $\mathbb{C}^n$. Let $1\leq i<n/2$. Consider the Radon transform between space of functions ...
7
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1answer
243 views

Kinematic formula for Euler characteristic

Is there a formula for $\int \chi(K \cap gL) \: dg$ (where $\chi$ is Euler characteristic) analogous to the kinematic formula for $\int \mu(K \cap gL) \: dg$ (where $\mu$ is Lebesgue measure)? In both ...
2
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0answers
111 views

Limit involving singular kernel: $\lim_{s\to 1}(1-s)\int_{\Omega}\frac{(u(x)-u(y))}{|x-y|^{d+2s}} d y. $

Let $\Omega\subset \Bbb R^d$ be a bounded $C^1$ domain. Let $u:\Bbb R^d\to \Bbb R$ be a function in $C^2_b(\Bbb R^d)$. I would like to compute the following limit: for $x\in \partial \Omega$ $$L= \...
3
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1answer
137 views

Mean cross-sectional area

A convex compact body $K$ in 3-space has well-defined volume, surface area, and mean width. Do these quantities enable one to say anything about the "mean cross-sectional area"? I put the phrase in ...
6
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0answers
122 views

Integral geometry for general closed smooth manifolds

Let $M$ be a closed smooth manifold of dimension $2n$ for some positive integer $n$. Let $\mathit{Diff}(M)$ be the group of diffeomorphisms of $M$. Let $L$ be a closed embedded $n$-dimensional ...
2
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0answers
29 views

Funk transform of density supported on an embedded curve

A Funk transform is a certain invertible linear transformation on the space of square-integrable functions on $S^2$. I think its domain can be extended to include densities supported on embedded ...
5
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3answers
141 views

Reconstructing a curve in $S^2$ from intersections with great circles

Take $S^2$ with its standard metric. The space of great circles in $S^2$ can be identified with the real projective plane $\mathbb{R}P^2$. Let $X$ be an embedded circle in $S^2$; associate to it a ...
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0answers
54 views

Kernel of Radon transform in $\mathbb{R}^3$

Consider the Radon transform from the space of functions on the manifold of affine lines in $\mathbb{R}^3$ to functions on the manifold of affine 2-planes in $\mathbb{R}^3$: $$(Rf)(H):=\int_{l\subset ...
3
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135 views

A problem on real analysis related to Hilbert's fourth problem

This is an extensive re-write of a question I deleted and which had basically the same title. Identify the cylinder $S^1 \times \mathbb{R}$ with the space of (co)oriented lines in the plane by ...
2
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2answers
107 views

Does this formula for caliper diameter hold for concave polyhedra?

I recently asked on MathOverflow and also asked several people I know to prove the following: How do I prove that the average caliper diameter of the polyhedron across all possible rotations is ...
7
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0answers
225 views

Integral representations of finite groups and lattice point geometry

This contains both a reference request, and a specific problem. Let $K$ be a finite group, and let $\theta: K \to {\rm GL}(d,{\bf Z})$ be a (faithful) group representation over the integers. Consider ...
5
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1answer
395 views

Convergence in the proof of Crofton's Formula

Let $\mathcal{L}$ be the set of oriented lines in $\mathbb{R}^2$ and let $\mu$ be the Kinematic Measure on $\mathcal{L}$; up to scaling, $\mu$ comes from the unique (up to scaling) volume form on $\...
3
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1answer
97 views

Under what hypothesis on the domain is the X-ray transform/John transform operator bounded?

I asked this question on math stackexchange, without any reply yet. Link:https://math.stackexchange.com/questions/1401580/under-what-hypothesis-is-the-x-ray-transform-john-transform-operator-bounded ...
6
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1answer
179 views

Radon transform between affine grassmannians

Let $\overline{GR}(n,k)$ be the manifold of all affine k-dimensional subspaces in $R^n$, and let $R:C^{\infty}_c(\overline{GR}(n,k))\to C^{\infty}_c(\overline{GR}(n,l))$, $0\le k<l\le n-1$, be the ...
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0answers
82 views

Combining microlocal Helgason's support and Holmgren's theorem to prove injectivity of limited-angle Radon transform

This questions is slightly related to Kashiwara's watermelon theorem and Microlocal version of Helgason's (support) and Holmgren's theorems, in which I asked for some references. Now I ...
4
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1answer
411 views

Kashiwara's watermelon theorem and Microlocal version of Helgason's (support) and Holmgren's theorems

I would like to find good references for the theorems mentioned above in the title. I am reading chapter VIII of Hörmander's classic, but I wonder whether there is something more up-to-date. My ...
3
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1answer
337 views

Kernel of the Radon transform

Consider the following generalized version of the Radon transform. Let $X,Y,Z$ be compact smooth manifolds. Let $p\colon Z\to X$, $q\colon Z\to Y$ be smooth maps. Let $m$ be a fixed smooth density (...
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66 views

Integral representation formula for convex

For $u \in \mathbb{S}^{d-1} \subset \mathbb{R}^d$, it is easy to show that: \begin{equation} u=c_d \int_{\mathbb{S^{d-1}}} \xi \mathbb{1}_{\left\{x \cdot u >0 \right\}}(\xi) \ \rm{d}\sigma_{d-1}(\...
3
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1answer
101 views

General Radon-type inverse problem

Let $f : \mathbb R^n \to \mathbb R$ be a density which is sufficiently smooth and can also be restricted to have compact support for now. Let $t \ge 0$ and $F_t : \mathbb R^n \to \mathbb R$, i.e. $(...
4
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1answer
218 views

Invertibility of an inverse problem

Let $p$ be a scalar field $p: \mathbb R^n \to \mathbb R$. I encountered the problem of reconstructing an unknown density $p$ from its integral values $$I(t,z) = \int_{V_t} p(x) dS$$ along a one-...
3
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0answers
111 views

Probability that a random projection doesn't reduce the distance of a point from a subspace too much

Consider the natural uniform measure (is it called the Haar measure?) on the set of $(n-k)$-dimensional subspaces of $R^n$. We are given a $d$-dimensional affine subspace $U$ (think of $d, k \ll n$; ...
24
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2answers
1k views

Random points on the unit sphere

Suppose you have $n$ points picked uniformly at random on the surface of $\mathbb{S}^d,$ and let the volume of the convex hull of these points be $V_{n, d}.$ Clearly, $V_{n, d}$ converges to the ...
7
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1answer
722 views

Idea and intuition behind Penrose transform

I would like to know what a Penrose transform is, or more precisely, what is it intended to be - I'm interested in ideas, intuition and some examples of application. My knowledge of differential ...
5
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1answer
243 views

boundary density of the Von Koch flake

Given a measurable set $K\subset \mathbb{R}^d$ we consider the occupation ratio $$f_r(x)=vol(K\cap B(x,r))/r^d$$ and especially the asymptotics when $r\to 0$. When $K$ has a fractal boundary and $x$ ...
4
votes
2answers
192 views

Reconstructing set of points from one-dimensional images

Consider a set of $N$ points in $n$-dimensional space, i.e. \begin{align*} \{x_1, \dots, x_N\} \subset \mathbb R^n. \end{align*} Let us be given a finite family of non-injective matrices \begin{...
3
votes
1answer
179 views

Partial recovery from Radon transform

Let $f : \mathbb R^3 \to \mathbb R$ be an integrable function. Let $\eta$ be a one-dimensional subspace of $\mathbb R^3$. We denote $p+\eta$ the affine subspace (a line) which is obtained by ...
3
votes
1answer
126 views

Generalized Radon transform (Relaxed sufficient condition for invertibility)

The generalized Radon transform maps a function $f \in L^1(\mathbb R^n)$, usually interpreted as a density of an object, to its integral value over an $(n-1)$-dimensional affine subspace. To be more ...
2
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1answer
229 views

“Limited angle” in n-dimensional Radon transform?

The Radon transform in two-dimensions is well studied. It maps a sufficiently nice function $f: \mathbb R^2 \to \mathbb R$ to its line integral along a certain line $L$, i.e. \begin{align*} Rf(L)...
4
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1answer
226 views

Interpretation of the integral “with respect to a plane wave” in terms of Radon transform

This question might have a formulation in higher dimensions, but for now let's deal with the 2 dimensional Radon transform: $\newcommand{\R}{\mathbb{R}}$ $$ Rf(\varphi,s)=\int_{-\infty}^\infty f(s\...
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0answers
92 views

A question on Radon transform

In the book of Helgason which is called "Radon transform and integral geometry", he defines on page 2 the Radon transform on hyperplanes as: $$ \hat{f}(\xi) = \int_{\xi} f(x) dm(x)$$ Where $dm(x)$ ...
8
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3answers
666 views

Rate of growth of an explicit integral

Let $$J_1=\int_0^1\frac{1}{\sqrt{1-t_2}}dt_2,$$ $$J_2=\int_0^1 \int_0^{t_2}\frac{1}{\sqrt{1-t_2}}(\frac{1}{\sqrt{1-t_3}}+\frac{1}{\sqrt{t_2-t_3}})dt_3dt_2,$$ $J_3=\int_0^1 \int_0^{t_2}\int_0^{t_3}\...
5
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2answers
556 views

Pseudo-Differentialforms

I'm looking for a definition of pseudo differential forms in ordinary differential geometry. However searching the web gave me nothing. There are definitions in supergeometry but that is not what I'm ...
7
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1answer
425 views

Inversion of Radon transform by incomplete data: specific case

Let $R[f](p,t)$ denote the Radon transform of smooth function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ with compact support in $\mathbb{R}^n_+$: $$ R[f](p,t) = \int\limits_{x \cdot p = t} f(x) dx. ...
3
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3answers
283 views

Reference wanted for application of Parametric Transversality

Let $\hbox{Aff(}k,n)$ be the space of $k$-dimensional affine subspaces of $R^n$. The group of Euclidean isometries of $R^n$ (the semi-direct product of rotations and translation) acts transitively on ...
3
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1answer
426 views

On the generalized Radon transform and currents

Given a family of hypersurfaces $H_{t,p} = $ {$x \in \mathbb{R}^n \mid g(x,p) = t $} one defines a generalized Radon transform $R$ of a function $u \colon \mathbb{R}^n \to \mathbb{R}$ as $$ R[u] (t,...
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2answers
864 views

Isoperimetric-like inequality for non-connected sets

The classical isoperimetric inequality can be stated as follows: if $A$ and $B$ are sets in the plane with the same area, and if $B$ is a disk, then the perimeter of $A$ is larger than the perimeter ...
2
votes
3answers
962 views

Draw a Random Line Through a Voronoi Tessellation, What is the Average Number of Voronoi Cell the Line Intersects?

Update: problem reformulation Following the advice in comments, I now restate my problem using Voronoi tessellation. Given a unit hypercube $H_n=\{(x_1,\ldots,x_n)\in \mathbb{R}^n: 0\leq x_i\leq 1\}$...
9
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2answers
705 views

The relationship between Crofton formula and Radon transform.

The famous Crofton formula says that the length of a curve can be calculated by integral of the `line crossing' over the space of all oriented lines. My question is, is there a way to treat this ...
53
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3answers
6k views

cube + cube + cube = cube

The following identity is a bit isolated in the arithmetic of natural integers $$3^3+4^3+5^3=6^3.$$ Let $K_6$ be a cube whose side has length $6$. We view it as the union of $216$ elementary unit ...
24
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4answers
2k views

Ellipse naturally associated with a polygon

My colleagues and I have stumbled onto a way to associate an ellipse, or equivalently a positive definite symmetric matrix, to a polygon that is different from other better known ways. We want to know ...
3
votes
2answers
514 views

If $K$ and $L$ are compact convex sets with smooth boundary, does their union have piecewise-smooth boundary?

Clarification: by "piecewise", I mean a finite number of pieces. I'm sure this must be true, but my search for a citation was in vain (although I did learn the new term "polyconvex"). Thanks!
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2answers
300 views

What are sufficient conditions on $f, g : \mathbb{R}^n \to \mathbb{R}$ such that `$\{x : f(x) \geq t \} \cup \{ x : g(x) \geq u\}$` has piecewise-smooth boundary?

What are sufficient conditions on $f, g : \mathbb{R}^n \to \mathbb{R}$ such that $\{x : f(x) \geq t \} \cup \{ x : g(x) \geq u\}$ has piecewise-smooth boundary? Some remarks: I don't mind if the ...
0
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1answer
164 views

Upper bound on the number of intersections of algebraic manifolds with affine planes

Given an algebraic map $f: B^d \to \mathbb{R}$, from the unit ball of dimension $d$ to the real, let $Y = f^{-1}(0)$. Then it is always possible to find a smaller ball $B_r \subset B^d$ not ...