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Questions tagged [integral-geometry]

Integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space.

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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 ...
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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 ...
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182 views

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 ...
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1answer
187 views

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 ...
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Derivation of The Cauchy-Crofton Formula on a plane.

I am currently reading an article. It proposes a proof of the Cauchy-Crofton Formula. There are some steps and concepts which I am not completely sure I understand. (I am also aware of another ...
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1answer
127 views

Cauchy-Crofton formula for hyperbolic disc.

I am currently reading the article Computing geodesics and minimal surfaces via graph cuts.I have difficulties with understanding the Cauchy-Crofton formula posed for 2D Riemannian space with ...
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32 views

Estimate of the length of a curve via average number of intersection with spheres

This is a problem in integral geometry. I can't prove or find a proof of the following inequality : Let $\beta$ be a simple closed curve and $Box(r)=\{|x|< r, |y|< 2^{1/3}r, |z|< 2^{2/3}r\}\...
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5answers
132 views

Calculate volume between two geometric figures

I have a figure C that is defined as the intersection between the sphere $x^2+y^2+z^2 \le 1 $ and the cyllinder $x^2+y^2 \le \frac{1}{4}$. How should i calculate the volume of this figure?
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1answer
177 views

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 $...
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1answer
654 views

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 $...
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1answer
199 views

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 ...
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2answers
56 views

Show that a curve which is perpendicular to a vector exactly twice is the unknot

If I know that the tangent vector $T(s)$ of $\gamma(s)$, a smooth closed curve in $\mathbb{R}^3$, is perpendicular to some vector $\vec{v}$ exactly twice, say at $s_0$ and $s_1$, how can I show that $\...
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1answer
763 views

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'...
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1answer
72 views

Proving that a proposed function is a Borel measure

Suppose $K$ is a fixed compact convex subset of $\mathbb{R}^n$. I wish to define a measure $M(K,\cdot):\{Borel subsets of \mathbb{R}^n\} \to \mathbb{R}$ where intuitively $M(K,A)$ (where $A$ is a ...
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1answer
217 views

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 ...
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Center of mass in a straight rod [duplicate]

I got an assignment to prove that in a straight homogeneous rod, you can always choose a coordinate system in such a way that $\int_S x_1dx_1dx_2=0 $ $\int_S x_2dx_1dx_2=0 $ $\int_S x_1x_2 ...
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1answer
93 views

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 ...
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282 views

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 ...
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1answer
2k views

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 ...