Questions tagged [integral-geometry]

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

19 questions
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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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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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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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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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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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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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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 ...