All Questions
7 questions with no upvoted or accepted answers
17
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0
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731
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Does every connected set that is not a line segment cross some dyadic square?
A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...
- 171
10
votes
0
answers
1k
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Interpolating points with minimum curvature constraint
I have $n$ points $p_i$ strictly interior to a rectangle $R$,
and I would like to connect them with a curve $C$ whose curvature is as low as possible.
Let $\kappa_\max(C)$ be the sharpest (largest ...
- 151k
5
votes
0
answers
235
views
Arrangement of points, lines, and planes
Is it possible to construct a finite nontrivial arrangement of points, lines, and planes in 3-dimensional Euclidean space with the following properties?
every line is incident with four points and ...
- 2,773
5
votes
0
answers
313
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Trade-off between covering number, ball radius and diameter of $d$-dimensional shapes
Given any $d$-dimensional shape $X$ in the Euclidean space, let $\ell(X)$ be the length of the longest line segment connecting two points of $X$. How can we prove the following statement?
There exists ...
- 1,677
3
votes
0
answers
60
views
A canonical map from a Euclidean cone-manifold $M^3$ to $\mathbb{E}^3/\mathrm{Hol}(M)$
Suppose we have a 3-dimensional Euclidean cone-manifold $M$—in my book that just means $M$ is a manifold whose geometry is constructed by gluing it out of Euclidean tetrahedra, with faces paired by ...
- 458
3
votes
0
answers
214
views
Volume of intersection of a ball and cube with arbitrary position in $n$ dimension
Let $ A(n, r, x) = B^n_r(x) \cap [0,1]^n $ denote the intersection between an $n$ ball $B^n_r(x)$ with arbitrary radius $r$ and arbitrary center $x \in \mathbb{R}^n$ that intersects a unit $n$ cube $ [...
- 365
0
votes
0
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106
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Upper bounds for minimum angle
What are the latest and best results on the asymptotic upper bound for the minimum angle between any pair of rays among $n$ rays in $\mathbb{R}^3$?
Any helpful answer would be appreciated. Thank you!
- 61