Questions tagged [integral-geometry]

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54 votes
3 answers
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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 ...
Denis Serre's user avatar
  • 51.5k
26 votes
2 answers
2k 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 ...
Igor Rivin's user avatar
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25 votes
4 answers
3k 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 ...
Deane Yang's user avatar
  • 26.9k
17 votes
2 answers
945 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 ...
Guillaume Aubrun's user avatar
9 votes
2 answers
848 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 ...
David's user avatar
  • 143
9 votes
1 answer
1k 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 ...
Ennar's user avatar
  • 193
8 votes
3 answers
700 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}\...
Fantastic's user avatar
  • 113
8 votes
0 answers
270 views

Integral representations of finite groups and lattice point geometry

See the edit at the bottom (April 2021) 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 ...
David Handelman's user avatar
7 votes
1 answer
287 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 ...
James Propp's user avatar
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7 votes
1 answer
143 views

Crofton formula: expected intersections is to length as variance is to what?

There is this beautiful Crofton formula for the length $L(C)$ of a curve $C$ on the round unit 2-sphere: you take the expected number of intersections of $C$ with a random great circle and multiply by ...
Jonny Evans's user avatar
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7 votes
1 answer
464 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. ...
Appliqué's user avatar
  • 1,269
6 votes
1 answer
192 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 ...
Dmitry Ryabogin's user avatar
6 votes
3 answers
184 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 ...
guest_1213's user avatar
6 votes
0 answers
130 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 ...
rori's user avatar
  • 257
5 votes
2 answers
1k 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 ...
Nevermind's user avatar
  • 624
5 votes
1 answer
596 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 $\...
Linda's user avatar
  • 51
5 votes
1 answer
635 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 ...
Qwertuy's user avatar
  • 251
5 votes
1 answer
290 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 $x$ ...
kaleidoscop's user avatar
  • 1,268
5 votes
2 answers
203 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{...
user avatar
4 votes
1 answer
252 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\...
icurays1's user avatar
  • 203
4 votes
1 answer
135 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 ...
Learning math's user avatar
4 votes
1 answer
232 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-...
user avatar
3 votes
2 answers
594 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!
Ryan O'Donnell's user avatar
3 votes
3 answers
333 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 ...
Dick Palais's user avatar
  • 15.2k
3 votes
1 answer
459 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,...
Appliqué's user avatar
  • 1,269
3 votes
1 answer
241 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 ...
James Propp's user avatar
  • 19.4k
3 votes
1 answer
118 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. $(...
user avatar
3 votes
1 answer
209 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 ...
user avatar
3 votes
1 answer
160 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 ...
user avatar
3 votes
1 answer
461 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 (...
asv's user avatar
  • 21.1k
3 votes
0 answers
103 views

Continuous, unitarily, translation invariant valuations

By a theorem of Alesker's, (Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations, J. Diff. Geom. 63 (2003, 63-95), the space $\text{Val}_n^{\text{U}(n)}(...
James Silipo's user avatar
3 votes
0 answers
114 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 ...
asv's user avatar
  • 21.1k
3 votes
0 answers
148 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 ...
alvarezpaiva's user avatar
  • 13.2k
3 votes
0 answers
119 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 $d, k \ll n$; ...
Navin Goyal's user avatar
2 votes
3 answers
1k 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 $H_n=\{(x_1,\ldots,x_n)\in \mathbb{R}^n: 0\leq x_i\leq 1\}$...
2 votes
1 answer
260 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)...
zeno44's user avatar
  • 41
2 votes
1 answer
227 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 ...
phaedo's user avatar
  • 123
2 votes
2 answers
136 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 ...
JDoe2's user avatar
  • 101
2 votes
0 answers
116 views

Integral geometric meaning of diameter

Let $X\subset \mathbb CP^n, n>2$ be a complex smooth algebraic hypersurface. Any hyperplane section $H\cap X$ is connected and has diameter $Diam(H\cap X)$ in the inner metric induced from the ...
Dmitrii Korshunov's user avatar
2 votes
0 answers
71 views

A regularity type question in integral geometry

The affine projective space $AP^1$ is the 3 dimensional analytic manifold consists of all lines passing through arbitrary points of the plane. For an smoith function $f:\mathbb{R}^2\to \mathbb{R}$ ...
Ali Taghavi's user avatar
2 votes
0 answers
108 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 ...
Vlad the magnificent's user avatar
2 votes
0 answers
128 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= \...
Guy Fsone's user avatar
  • 1,033
2 votes
0 answers
31 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 ...
man's user avatar
  • 305
2 votes
0 answers
106 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 ...
Qwertuy's user avatar
  • 251
1 vote
2 answers
329 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 ...
Ryan O'Donnell's user avatar
1 vote
0 answers
237 views

A sort of dual to nondegenerate random variables

I was motivated by this classical puzzle/1992 Putnam problem. Suppose 4 points are independently and uniformly distributed on a sphere in 3d. What is the probability the tetrahedron they form contains ...
Jess Boling's user avatar
1 vote
0 answers
80 views

Closed form volumes for intersecting modified cylinders

This question is somewhat related to the question Intersecting cylinders, but where the cylinders are now modified to orbifolds in the hypercube with singularities occurring at the vertices of the ...
53Demonslayer's user avatar
1 vote
0 answers
31 views

Helgason's support theorem type result in 2 dimensions

I had posted this question in math stackexchange here. Let $\Omega \subset \mathbb{R}^2$ be an open domain with smooth boundary. Identifying $\mathbb{R}^2$ with $\mathbb{C},$ consider the following ...
Rahul Raju Pattar's user avatar
1 vote
0 answers
70 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 ...
asv's user avatar
  • 21.1k
0 votes
1 answer
172 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 ...
John Jiang's user avatar
  • 4,354