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40
votes
3answers
4k views

cube + cube + cube = cube

The following identity is a bit isolated in the arithmetics 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 ...
22
votes
2answers
576 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 ...
21
votes
4answers
1k 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 ...
12
votes
2answers
636 views

Isoperimetric-like inequality for non-convex 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 ...
8
votes
3answers
415 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 ...
8
votes
2answers
367 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 ...
7
votes
1answer
341 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. ...
6
votes
1answer
247 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 ...
4
votes
2answers
172 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 ...
4
votes
1answer
112 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 ...
3
votes
2answers
207 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 ...
3
votes
1answer
377 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] ...
3
votes
1answer
133 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
79 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 ...
3
votes
1answer
50 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. ...
3
votes
1answer
113 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 $x$ ...
3
votes
2answers
106 views

Reconstructing density from integrals along specific manifolds

Let $\Phi_t : \mathbb R^n \to \mathbb R^n$ be the time-$t$-map associated to an ODE $\dot{x}=F(x)$ and let $H: \mathbb R^n \to \mathbb R$. Let $F$ and $H$ be sufficiently smooth (e.g. $C^k$ or ...
3
votes
0answers
61 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 $d, k \ll n$; ...
2
votes
2answers
335 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!
2
votes
2answers
199 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 ...
2
votes
3answers
827 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 ...
2
votes
1answer
111 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*} ...
1
vote
2answers
256 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
votes
1answer
132 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 ...
0
votes
0answers
42 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) \ ...
0
votes
0answers
55 views

Existence and Uniqueness of Volterra integral equations of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To prove the existence and uniqueness of solutions of Volterra integral equation(VIE) of the first kind, we usually differentiate it and convert to the VIE of the ...
0
votes
0answers
61 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 $dm(x)$ ...