Integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space.
Questions tagged [integral-geometry]
34 questions
20
votes
1 answer
Why isn't differential Galois theory widely used?
Ellis Kolchin developed differential Galois theory in the 1950s. It seems to be a powerful tool that can decide the solvability and the form of the solutions to a given differential equation.
Why isn't differential Galois theory widely used in…
Henry.L
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10
votes
2 answers
Simple proof of the Cauchy-Crofton formula on the sphere?
Let $\gamma$ be a regular curve on the sphere. In a lecture, the following result was used
$$L(\gamma)=\frac 14 \int_{S^2} \sharp (\gamma \cap \xi ^\perp)d\xi$$
$\xi^\perp$ is the plane with normal $\xi$ going through the origin. $\sharp(\gamma\cap…
user153312
9
votes
1 answer
Continuity of the Euler characteristic with respect to the Hausdorff metric
Hadwiger's theorem of integral geometry states that all continuous valuations which are invariant under rigid motions are expressible in terms of the intrinsic volumes. The continuity property means with respect to the Hausdorff distance:
$$d_H(X,…
7
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1 answer
Average Diameter of a polyhedron
Define the caliper diameter of a polyhedron as follows:
Let $P_1$ and $P_2$ be two planes both of which are parallel to the x axis such that the perpendicular distance between $P_1$ and $P_2$ is the smallest possible distance allowing the whole of…
JDoe2
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0 answers
How did Steiner prove his famous formula?
In convex integral geometry and geometric measure theory, Steiner's formula is the name of the following elegant result:
Let $B_n$ be the unit ball in $\mathbf R^n$. If $S$ is a nonempty bounded convex subset of $\mathbf R^n$, then for…
Cave Johnson
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6
votes
1 answer
Inverting Fourier transform "on circles"
Dear Math enthusiasts,
I am struggeling with a problem for which a solution is already given to me, but I can just not see why it is true. Here is the setting: I am given a function $f(x,y,t)$. It's well behaved let's say. Smooth and things like…
Florian
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5
votes
1 answer
The Cauchy-Crofton Formula
I am trying to understand the most basic formula from integral geometry. I have been looking at this website.
The problem is things aren't working out for very simple examples. The circle works fine. It has perimeter $2\pi$ and every line…
Math Student
- 709
5
votes
0 answers
Integral Geometry Reference Request
I am looking for a good introductory reference (book, lecture notes, survey article) on integral geometry. I am especially interested in the Crofton formula in $\mathbb{R}^n$ and its extensions to Riemannian manifolds.
So far, I have only had a look…
begeistzwerst
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4
votes
1 answer
Surface area of Convex bodies contained in one another
Suppose we have two compact convex bodies one contained in the other in $\mathbb{R}^n$, $C\subset D\subset \mathbb{R}^n$. Does it follow that the ($n-1$ dimensional) surface area of $C$ is less than $D$? If so is there a natural sequence of $n-k$ …
Josh F
- 481
4
votes
1 answer
What is the average width of a given tetrahedron?
I have a tetrahedron with (1, 1, 1), (2, 1, 1), (1, 2, 1), (1, 1, 2) vertex. What is the average width? I don`t know how to start it. I need to find a useful parameterization. Please help me with any ideas.
Bea
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3
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1 answer
A question from Selected Topics in Integral Geometry
I'm referring to Gel'fand, Gindikin, and Graev's Selected Topics in Integral Geometry, pages 4-5, section 1.4 (see here and here).
Now in page 5 they write that:
$$dx_1 dx_2 = d(\xi_1 x_1 +\xi_2 x_2 -p) d\mu_{\xi}$$
where…
MathematicalPhysicist
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2
votes
1 answer
Radon Transformation
I have been tinkering over the line segmentation of images. I found that it is very well implemented in matlab with the Hough algorithm.
Now the Hough-transformation is just a special form of the Radon Transformation.
I was now wondering how can…
A.Dumas
- 253
2
votes
0 answers
Derivation of Crofton's formula in $\mathbb{R}^4$
I am having trouble deriving Crofton's formula for rectifiable curves in $\mathbb{R}^4$. My approach so far has been as follows. Hyperplanes in $\mathbb{R}^4$ may be given by the data of a real number $r$ and a point $v$ on $S^3$ via…
strtlmp
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2
votes
0 answers
Buffon's needle with nonunifom distribution of drops
The famous Buffon's needle problem relates the area of a region $C$ (or length of a curve $C$, in 1D) to the expected number of times a randomly dropped needle will fall on the region.
This makes sense if the lengths of all needles are the same, and…
blue_egg
- 2,333
2
votes
0 answers
CDF of the distance from origin to the hyperplane passing through $d$ i.i.d. points in $\mathbb{R}^d$
I am stuck with a problem in multivariable statistics. The problem can be stated as follows:
For a spherically symmetric distribution in $\mathbb{R}^d$, it can be specified completely by the function $F(r)=\Pr(\|X\|>r)$. For example, 2d standard…
zhaofeng-shu33
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