All Questions
718 questions
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Optimal fitting of spheres in a cylinder
How to find the minimum height and width of a cylinder containing n identical spheres?
- 21
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85
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More on triangles inscribed in convex regions with one vertex fixed
We add a bit to On maximum perimeter triangles inscribed in convex regions with one vertex fixed. Let C be a convex planar region and P a point on its boundary.
Are there convex shapes C other than (...
- 5,979
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320
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A formula for the area of bicentric quadrilateral
Can you provide a proof for the claim given below? The following claim is inspired by Harcourt's theorem and can be seen as its generalization to quadrilaterals.
Claim. Given bicentric quadrilateral $...
- 2,661
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416
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A metric geometry problem which calculates the limitation of human eyes
This is the update version of this question A functional inequality which calculates the limitation of human eyes
Let an Euclidean space $M$ (or a path connected metric space) be partitioned into ...
- 1,462
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124
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A center of convex planar regions based on chords
This is based on Chapter 6 of 'Convex figures' by Yaglom and Boltyanskii. This post also continues On two centers of convex regions.
A point $P$ in the interior of a planar convex region $C$ divides ...
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69
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Holder-continuous barycenter maps
Let $(X,d)$ be a complete locally-compact metric space. We define the $p$-barycenter map as a continuous function:
$$
\beta:\mathcal{P}_p(X)\rightarrow X,
$$
which is a right-inverse of the map ...
- 5,405
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49
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When can a compact metric space be covered by finitely many nearly-disjoint closed and convex sets?
This question is a follow-up of the following negative question.
Let $(X,d)$ be a (non-empty) compact metric space.
More generally than in the first post, I'll call a set of non-empty subsets $C_1,\...
- 5,405
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98
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Asymptotic optimal sphericity
How quickly does maximum sphericity of polyhedra with $n$ faces approach 1 as $n→∞$? I can show that sphericity $1 - \frac{5 \sqrt{3} π}{27n} - O(n^{-3/2})$ is possible. Is this, especially $O(n^{-3/...
- 7,545
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47
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Cutting convex regions into equal diameter and equal least width pieces - 3
We add a bit to Cutting convex regions into equal diameter and equal least width pieces - 2. There, we asked, for example: If we divide a 2D convex region C into n convex pieces such that the maximum ...
- 5,979
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936
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The exact number of points within a circle of radius r centered on a lattice point in a hexagonal lattice? Review expression Gauss circle problem
In the case of a square lattice, the exact number of points within a circle of radius r centered in the center is (see: http://mathworld.wolfram.com/GausssCircleProblem.html:
$$N(r)=1+4Floor(r)+4 \...
- 31
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113
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How much a probability distribution is non-uniform in a convex subspace of $\mathbb{R}^d$?
I know a number of (standard and well known) ways to measure the distance between two probability distributions and, more in general, to quantify how much one is far from another.
Could you please ...
- 1,677
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4
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457
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Confining a polytope to one side of an affine hyperplane
Judging whether one convex polytope is inside of another when both are expressed as a system of linear inequalities seems not to be an easy problem.
This answer on math.stackexchange.com claims the ...
- 2,239
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93
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On smallest convex m-gons that contain a given n-gon where m<n
Given a convex n-gon region P, and an m less than n, will the least area convex m-gon Q that contains P be such that an edge of Q coincides with an edge of P (in other words Q cannot be such that P ...
- 5,979
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189
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Terminology "upper" Ahlfors regular measure
Let $(X,d)$ be a metric space and $m$ be a Borel measure on $(X,d)$. The measure $m$ is called Ahlors regular if $m(B(x,r))\asymp r^q$ for some $q>0$ and each $x\in X$. Is there a name for ...
- 5,405
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357
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Metric space that is not a subspace of $\mathbb{R}^n$? [closed]
What are some simple examples of metric spaces that cannot be subspaces of $\mathbb{R}^n$? I've heard there is an example with $4$ points, where two points lie between the other two, but I cannot ...
- 1,465
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42
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Construct pairs of $n$-dimensional convex bodies with given ratios ($p$) of volumes
Given a dimension $n$ and a number $p \in (0,1)$, to what extent is it possible (in what cases) to construct a convex set $A$--not a hypersphere--and a "snugly" inscribed (InscribedFigure) ...
- 723
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89
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What is the minimal number of lines needed to partition a simplex into cells of diameter at most $\epsilon$?
I am studying a problem that requires me to partition the simplex into cells using a particular family of hyperplanes. For concreteness, consider the 2-simplex. I would like to construct lines ...
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127
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What does it mean exactly for a pair of $S^0$'s to be unlinked on a knot $K$?
I am trying to learn about the effects of knot mutation on the hyperbolic manifolds obtained via hyperbolic Dehn surgery, and I'm currently reading Ruberman's paper "Mutations and Volumes in $S^3$" (...
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