All Questions
Tagged with integral-geometry convex-geometry
8 questions
1
vote
0
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245
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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 ...
- 646
1
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0
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83
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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 ...
- 383
7
votes
1
answer
302
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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 ...
- 19.7k
3
votes
1
answer
262
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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 ...
- 19.7k
2
votes
2
answers
137
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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 ...
- 101
0
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0
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73
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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) \ \rm{d}\sigma_{d-1}(\...
2
votes
3
answers
1k
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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 $H_n=\{(x_1,\ldots,x_n)\in \mathbb{R}^n: 0\leq x_i\leq
1\}$...
25
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4
answers
3k
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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 ...
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